In a modern western perspective, the concepts of justice and law seem almost inextricably linked, with major philosophical debate around the exceptional times where they aren't (ie: is civil disobedience in the face of an unjust law acceptable), so it seems unusual to have separate gods for both concepts, especially with a much blurrier line than, say, Ares, god of War (read as: conflict and bloodshed) and Athena, goddess of War (read as: strategy and tactics).
I know there are few surviving mentions of Forseti, let alone myths, but is anything in their society or culture that might give a clue as to this split?
Well there are two problems you have going on here.
First, you assume that Forseti and Tyr were consistently considered patrons of these particular spheres of life in Norse religious life pre-conversion. You are absolutely right that Forseti is scarcely mentioned in the available literary evidence, and attempting to reconstruct him as deity of law of courts is likely to have little relation to any cultic worship of him, if any, among the Norse.
Norse deities, even the ones more extensively attested than Forseti, did not exist as static figures across the Norse world, both in regards to space as well as time. What worship of a figure like Tyr meant in Denmark in the 9th century would be different to how Tyr would be seen in say 11th century Iceland. Yet because 99% of popular understanding of Norse religion is filtered through the sagas, any cursory examination online will leave you with a very skewed idea of what Norse religious beliefs and practices actually were.
Secondly, you're importing modern conception of law and justice to a time frame that would not necessarily have made similar assumptions. Our modern conception of law is influenced by a host of institutions and beliefs, such as the existence of lawyers, the ideal of judicial impartiality and independence, secular law, and so on. These same assumptions need not, and indeed were not, applied in the Middle Ages.
If you're curious about the exercise of law in the early Middle Ages, I wrote an answer on law in Anglo-Saxon England that might be helpful as well
If you are curious I can go on at greater length about the political (and religious philosophy) inherent to Anglo-Saxon law in the 11th century.
However for the Norse specifically it is difficult to extrapolate to them as they left behind no legal texts prior to conversion and the sagas are inherently unreliable. Some legal codes contain pre-conversion elements within them, but these are mostly regarding punishments and fines, not legal theory or the distinctions between law and justice, and that ignores the problems with assuming the law codes to actually be indicative of legal practice on a day to day basis.
You might want to read this comment I've written earlier, trying to disentangle people's assumptions based on the Eddas, from what we know about the actual religion.
The idea that the gods really had specific functions in the way the Greek/Roman ones is rather hard to support, much less what those functions were. Adam of Bremen (who wasn't a first-hand account) refers to Odin as a god sacrificed to for fortune in war and against enemies, Thor as a weather and harvest god sacrificed to in times of famine, and Freyr as a fertility god. If you compare that to for instance the week-day names, which were 'translated' from Roman to Germanic deities, there, Thor is equivalent to Jupiter. But Jupiter as the 'allfather' would rather equate to how Snorri describes Odin. Odin in Adam's version would be more similar to Mars, but in week day terms Tyr deemed is equivalent to Mars. Freyr as a fertility god and equivalent of Venus is however fairly consistent.
Many of these attributions of the roles of gods you'll find confidently stated in popular-mythology texts are really uncertain and often based on shaky grounds. Gylfaginning, in Snorri's Edda, says about Ullr (and this is almost all it says about Ullr): Hann er bogmaðr svá góðr ok skíðfærr svá, at engi má við hann keppast. (He's a bowman and so good at skiing that none can compete with him) This is enough for many to have proclaimed Ullr the god of skiing and hunting. (nevermind that e.g. the giantess Skaði is described as a very good skier too) Snorri may not have known anything at all about Ullr beyond this information, which he might've inferred from some older kennings (by-names) for Ullr.
That niche would not seem to explain why there would be about as many place names in Sweden for Ullr as there are for Thor and Odin. And there are far more vi sites - a particular type of cult site - named for Ullr than those two together. Indeed the evidence from place names of the actual cult has fairly little to do with mythological prominence.
When it comes to Forseti, as I said in my previous comment, there's little evidence for him in Scandinavia. The only really definite account of a cult of Forseti is in Vita sancti Willibrordi , and refers to Frisia in the 700s. There is only one plausible place name in Scandinavia named for him; Forsetlund , attested from 1480 in Østfold in Norway.
Snorri recounts a poem from Grímnismál about Forseti. It says he sits all day in Gltinir, a hall with golden supports and a silver roof, and settles all disputes. (my paraphrase) Again, Snorri may not have known much about Forseti and extrapolated this role as judge-for-the-gods from the poem he cites as a source.
As with Ullr, this is enough for some to proclaim Forseti a god of justice or courts and such things. I don't think it is. If all you knew about the Christian god is that "he sits in heaven and judges" you might arrive at the same conclusion if you were trying to fit him into a pantheon. That's the thing - this presupposition that there was a Greek-like pantheon is uncertain. We don't know why those who worshipped Forseti, if there were any, did so. On the evidence we have, it doesn't seem like there was not have been many of them, the one mention of such a cult is a full 500 years before Snorri and it's entirely possible the cult of other gods had taken over by the end of paganism in Scandinavia. (the same thing may account for the absence of Ullr in the written record, despite an obvious widespread cult)
In many ways, Tyr is even more problematic. In terms of place names his cult was even more localized. I mentioned in that other post Gautatýr (Tyr of the Geats) in Skáldskaparmál as one of Odin's names, possibly indicating regional variation there. Which is only one of a half-dozen variants of Týr that Snorri mentions (Hangatýr, Farmatýr, Reiðartýr, Gautatýr, Hroftatýr, Sigtýr). This is one example of the name/term týr seemingly being used to simply mean 'god'. (Which would be consistent with its etymology; Proto-Germanic *tiwaz from Proto-Indo-European *deywós, 'god'; a cognate of Latin deus) There's ample evidence of cult of Tyr going back far in time, but it may have morphed with time and place. Despite being one of the 'older' and well-attested gods, he's a comparatively minor figure mythology-wise, but the fact that he has a weekday is enough to show the cult went way back in time, and there is ample evidence of his cult.
The suggestion Tyr was a justice god goes back to Dumézil; and like a lot of his theories, it's quite speculative and not believed by most today. Basically the whole status relies on interpreting that one myth as an example of the punishment for oath-breaking, and therefore justice? (it has been pointed out that it's not really congruent with a justice god to engage in perjury)
The other relevant mention is verse 38 of Lokasenna in the Poetic Edda (the story where Loki turns up at a party and insults everybody), where Loki says Þegi þú, Týr, þú kunnir aldregi bera tilt með tveim, handar innar hægri mun ek hinnar geta, er þér sleit Fenrir frá.". Freely translated:
Shut up Tyr/you could never/ "bera tilt" between two /your right hand/ I'd like to mention/the one Fenrir ripped off.
Whole articles (plural, see e.g. Sigurðsson) have been written on what the meaning of those four words "bera tilt með tveim" are; particularly "bera tilt", a frequent guess is "get two people to agree". The question is how to interpret it and how it might relate to the lines concerning his hand; perhaps it relates to oath-breaking rendering him unable to be trusted as a mediator. Or he's meant to be a mediator and "god of justice" but Loki is insulting him by implying he's not. The main focus though is linguistic though, just understanding what the idiom is here.
Personally I don't think Lokasenna is anywhere near well-written enough to allow to read in subtleties about Tyr's character. (it's commonly considered perhaps the weakest part of the Poetic Edda) There's no story in the verses; things only really happen and motives are explained in the prose prologue and epilogue tacked on. The rest is 65 verses of insults and threats hurled back and forth. Verse 37 is Tyr getting up and talking about how great Freyr is - apropos nothing as Freyr's not been mentioned earlier. Verse 38 is responding to Tyr but doesn't really have anything to do with what he just said. I think "Haha you've got no hand" as a mere random insult is consistent with the text.
But as said in that earlier post, this is skaldic poetry. The kind of thing to be performed at the kind of VIP banquets that it itself depicts. It's entertainment, not scripture, and we can't reliably infer anything about the religious roles of the gods from these stories. Much less take that a step further and try to infer things about Norse society from these roles.
Sources mentioned
Dumézil, Gods of the Ancient Northmen, ed. E. Haugen, Berkeley, 1973
Sigurðsson, “bera tilt med tveim : On Lokasenna 38 and the One-Handed Týr, Arkiv för nordisk filologi 121 (2006), pp. 139-160
