This may be highly irregular, but I managed to hunt down copies of the relevant scholarly sources and am able to answer my own question.

There are two sites being discussed here, Fara (Shuruppak) and Abu Salabikh, which are relatively near to each other. These sites are considered to be contemporary to each other and the tablets for both are often dated to 26th century BCE. Before radiocarbon dating the tablets at these sites were both dated based upon characteristics of the cuneiform written on the tablets. Robert Biggs describes the specifics of the method used for dating them in Inscriptions from Tell Abu Salabikh - Full text here, pp 24-26. The paragraphs below paraphrase.

The tablets from Tell Abu Salabikh and Fara share some characteristics that seem to point to them being contemporary to each other. Specifically, the two sites shared many of the same texts, with some of the tablets even sharing the same number of lines in the columns. The tablets at both sites use an almost identical set of archaic signs in the tablets and the type of colophones used on the tablets is also almost identical. Based upon these similarities, Biggs concluded that the tablets found in Abu Salabikh were approximately contemporary to the tablets found in Fara.

The Fara tablets were dated by the German Assyriologist Anton Deimel in the 1920s. He dated them as being from bit before the the reign of the king Ur-Nanse of Lagash (~2550 BCE) based also upon the characteristics of the writing itself. This was controversial at the time, with some scholars dating the tablets to the time of Urukagina, 150-200 years later. Deimel successfully defended his older dating in a series of articles published in the journal Oreintalia. After his 1928 article “Die zeitliche Ansetzung der Fara-Texte” no further challenges emerged and Deimel’s dating was generally accepted by the Assyriology community.

Another potential issue with this dating of the Fara tablets comes from the fact that the dating of Ur-Nanse’s reign was a bit muddy. Most of the texts from Lagash are considerably newer than the ones that Deimel was comparing against. While there was a clear list of rulers leading back to Ur-Nanse, the length of some of their reigns was unclear, making the dating of the tablets from the reign of Ur-Nanse similarly unclear.

Based upon Deimel’s dating of the Fara tablets and the similarities between the Fara tablets and the Abu Salabikh tablets and assuming that Ur-Nanse reigned around 2550 BCE, Biggs dates the Abu Salabikh tablets to around 2600 BCE.

Mentions of the Abu Salabikh tablets (including the Kesh Temple Hymn) dating to 2600 BCE typically reference Biggs as their source. Biggs assumed that these tablets were contemporary to the Fara tablets, which were dated by Deimel based upon characteristics of their writing compared to tablets from Lagash that were known to come from the reign of Ur-Nanse, the date of whose reign was unclear.

I seems like there would be a lot of room for error, but it turned out that date was actually pretty accurate.

If you read through it, Inscriptions from Tell Abu Salabikh contains a detailed description of the excavation site and notes a pretty good amount of organic matter found there including wood and bones. I did some further research and it turns out that some of this material from Area E was radiocarbon dated. Material taken from two different excavations of the site were radiocarbon dated by two different laboratories and the material was dated to between 2520 and 2550 BCE. The results from both labs agreed. (H. Wright, “Problems of Absolute Chronology in Protohistoric Messopotamia,” Paleoorient, vol 6, 1980, p 95 - Full text here ).

In summary, we're pretty confident that the tablets from Fara and Abu Salabikh (including the Kesh Temple Hymn and the Instructions of Shuruppak) come from some time between 2600-2500 BCE.

Edit: Clarified that Fara is an alternate name for Shuruppak