There's a lot to unpack here. First of all:

Julius Caesar repeatedly broke Roman law during his time as Consul

Did he? Caesar was a proconsul, not a consul (his consulship ended in 59), but setting that aside when did he break the law? Illegality of Caesar's actions in Gaul was never an especially important avenue of attack against Caesar. Only once was Caesar's activity in Gaul questioned, by Cato after Caesar attacked the Germans, apparently during a truce. Cato, says Plutarch, suggested that Caesar be turned over to the Germans for punishment, but nothing came of it and Cato was in the habit of taking hard, often outrageous, lines on things for which he was generally ignored. The "prosecution hypothesis" rests not on some perceived charge against Caesar for his time in Gaul but on the questionable legality of his legislation in 59. This legislation, it was asserted, had been passed by violence, although the violent incident was more or less manufactured by the actions of Bibulus, Cato, and three tribunes. A brief threat in 58 by P. Clodius to invalidate Caesar's legislation on auspicial grounds was abandoned, and there is no sign that illegality was ever raised again as a motive for recalling Caesar. In 56 when Domitius Ahenobarbus threatened to recall Caesar if elected consul, we are never told that his purpose was to prosecute him, merely to remove him from the province. It's a non-issue in the de provinciis consularibus. Plutarch fails to mention fear as a motivator for the conference at Luca, almost certainly anachronistically asserting that the triumvirs wanted to take over the state, and Suetonius says that Domitius' threat was that he would take away Caesar's armies, with no mention of prosecution (or even, explicitly, of recall from the province).

the Conservatives were very eager to convict him

I think I've already established that the textual evidence for a concerted attempt to prosecute Caesar in the 50s is quite slim at best. Caesar himself never mentions the threat of prosecution in the de bello civili. I've drawn out this quotation, however, because I think you're fundamentally misunderstanding something about the way the Republican state functioned. The dominant view in the study of the Republic following Gelzer and Syme was that there were no ideological issues at stake in Republican politics. Members of the senatorial class formed "factions" with each other, based on blood ties, amicitia, etc., but Gelzer and Syme argued that these should not be understood as anything like political parties, with set ideologies and structures, but rather as groupings of aristocrats setting themselves off from other aristocrats. This "frozen waste" theory has mostly been put to bed since the 80s or so, and the concept that Republican politics may well have been about real political issues and involved more than simply personal prestige and personal grudges has long gained traction in some form. Nevertheless, we see little sign of anything like a sorting of politicians or a self-identity along ideological lines that would line up with what we think of as "conservative" and "progressive." Presumably you're translating optimates as "Conservatives," but let's examine that a bit. The word optimas is exceedingly rare, although boni is rather more common in Cicero. Cicero seems to use it to designate everybody with the best interests of the Republic in mind, which to him generally means anybody who agrees with him. It's a terribly nebulous term that has gathered scholarly attention for the better part of two centuries. It seems to have fairly little to do with consistent personal ideology. Cato--arguably the optimas par excellence, after Cicero himself--famously proposed a grain bill during his tribunate in 62, and assisted P. Clodius in the funding of his grain bill by accepting an extraordinary commission in Cyprus. Cato is only the most famous example, but cases like this abound. One could drift between the spaces of popularis and optimas at different parts of your career, and most tribunes did so. Sallust has a comment about young men, presumably tribunes, causing agitation and stirring up the people. Sallust himself was just such a tribune, a novus homo who in his youth was politically active as a populist politician. Yet when he wrote his works he wore his moralist hat as an older man. There are a lot of reasons to suppose that populares tended to be younger men--not least because they historically were younger men, holding the tribunate, a junior office--and many (Tatum even goes so far as to claim all but two, which may be stretching it) populares came to be firmly within the ranks of the optimates once they reached the praetorship or consulship. This is unsurprising, really. Increasingly we see ideological baggage surrounding the tribunate's role as the mouthpiece of the people, whereas curule magistrates were decidedly more conceptually senatorial, interacting with them more than with the citizen body, whose powers of election and legislation were really the place of the tribunes. Meier's conception of popularis as a tactic is maybe not quite totally correct, in that it ignores the popular pressure under which so many tribunes acted, but it's on the right track. These certainly aren't political parties, and they're not necessarily terms associated with particular political "fields," a concept which, in the modern formulation, would not necessarily make a ton of sense to a Roman.

There are some other problems here--I'm unaware of any five-year limit on promagistrates, and I'm not sure what tribunician actions you see overturning professio and the limit of imperium at the pomerium (are you referring to the Law of the Ten Tribunes?)--but more importantly I think you're taking the wrong approach to the structure of the Republican state. Mommsen conceived of the Republic as a Rechsstaat. He observed the unification of Germany during his own lifetime and the establishment of clearly defined constitutional and legislative structures and more or less applied this to the Roman state. I don't think there's anybody still alive who would agree with this these days. Gelzer and especially Syme were the first to take a shot at it, pointing out the prevalence of personal connections in Roman politics, which led to the "frozen waste" model, which is itself no longer really accepted. The political structure of the Republic, and maybe most noticeably the role and place of the magistrates evolved over time, in a more or less haphazard manner.

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