If anything I think Caesar and Pompey would try and use her as a marriage tool to designate a potential successor (if not in name then at least in practice) by giving her hand to a political ally (e.g. Publius Clodius, Quintus Cicero, or Domitius Ahenobarbus). I think the history of the late Republic shows that political equilibrium is impossible when multiple powerful men hold sway in Rome at the same time. Ever since the rivalry between the Gracchi and Scipio Nasica, there was never a stable political situation in Rome until after Actium. One possible exception is the relationship between Octavian and Agrippa, but I'd view this as the exception rather than the rule. Since Pompey and Caesar had independent bases of support (read: different legions and clients loyal to each other), it wouldn't be possible to reconcile them with something as simple as a daughter.
But for the sake of argument, let's assume that it does make a difference. By the time of Julia's death in 54 BCE, Crassus had left Rome and Caesar was busy in Britain, so Pompey was well-positioned to make the most of his situation. Cicero was back in Rome, and all our written records indicate that Pompey and Cicero were close allies for much of the period from 58 BCE to Pompey's death in 48 BCE. With a powerful ally like Cicero and a daughter as political capital, Pompey would likely attempt a marriage between this new daughter (Pompeia from here on out) and either Cicero's brother Quintus (in an attempt to win over his support from Caesar's camp), or a family relation of one of his other senatorial allies such as Metellus Scipio, Domitius Ahenobarbus, Marcus Bibulus, or even Marcus Brutus. Brutus would be an interesting pick since Brutus himself hated Pompey, but the marriage of Pompeia could be used as a tool to either expand his political clientele or to solidify the cohesion of his existing clientele. Assuming the Civil War was not inevitable, and that Pompey was not actively trying to undermine Caesar, Brutus or Quintus Cicero would be natural choices (or even Publius Crassus if his early death is avoided), since both men had good relationships with Caesar. The TL would unfold depending on which choice Pompey makes so I'll propose two solutions depending on which course of action he takes.
Pompey becomes sole ruler of Rome:
With a POD at 54 BCE, Caesar is at the current high point of his career (having "won victories" in Britain and Germany), but Pompey's star is also rising, given his consulship in the previous year and upcoming consulship in 51 BCE (assuming that Clodius dies at the same time). Even if events play out slightly differently, Pompey's urgent goal is to remove Publius Clodius and Titus Milo from power, and to that end, Pompey begins to raise troops in Italy, perhaps even bringing some over from Spain or Greece. The events between 54 and 50 BCE play out the same (Crassus dies at Carrhae, Caesar wins at Alesia, the senate orders Caesar to stand down his army). The real critical juncture comes in 49 BCE. The only difference between TTL and OTL is that Pompey has custody over Caesar's granddaughter. Pompey still marries Metellus Scipio's daughter ITTL, and therefore his ties to Caesar are minimal. If anything, Pompeia's status in Rome as Caesar's only descendant could jeopardize Caesar's own dynastic plans, so Pompey (or the senate if you like) is holding all the cards in this scenario (just like IOTL). Pompey has total control of the senate and Caesar has bribed the tribunes, so politics are broken. The only real variable here is what Caesar does when his proconsular term expires. If he views his granddaughter as vital to his future political plans, then he has virtually no choice but to step down from power, since she could be killed if any open violence breaks out. Of course, IOTL he had "no choice" but to step down, and he didn't, so it's possible that he may march on Rome ITTL regardless. Maybe Pompey wins some political points since Caesar is marching on Rome with his granddaughter in the city, "Can you believe that? What utter disregard for his own family! What a monster!" but ultimately probably very few effects from this development. So maybe Pompey is able to win over some marginal senators with promises of a future marriage to his daughter, but I have my doubts. In this new civil war, Caesar loses and is either executed or banished, leaving Pompey in sole control of Rome. Does he establish a dictatorship? Unlikely, since he was already powerful in the senate and had few rivals for power besides Caesar and Crassus, and they're both gone. Maybe Cato complains a bunch ITTL, but that wouldn't really be a stretch from IOTL. The Republic survives a bit longer, but eventually it's probable that the same rivalries will emerge one generation down the road (this time perhaps Sextus Pompey will fight against Brutus and Cassius), or the eventual husband of Pompeia will himself become a man of influence in opposition to the Catonians. Either way, the civil wars will likely continue, and eventually either a descendant of Pompey or another powerful demagogue is finally able to become emperor (or the Roman state falls apart).
Pompey creates the second triumvirate:
Same POD as above, except we assume that the senate is the manipulative faction, and Pompey holds no animosity towards or fear of Caesar. In this scenario, Pompey will marry Octavia (sister of Octavian) which is what Caesar offered IOTL after Julia's death. With their marriage ties reinforced, Pompey would be faced with a problem. Clodius was an ally of Caesar at the time (at least implicitly when Caesar supported Clodius' brother for the consulship in 54 BCE), and Pompey was marginally an ally of Cicero at the time (given Pompey's role in the ending of Cicero's banishment). However Clodius and Cicero were bitter enemies, and they were both powerful men in the city of Rome. So Pompey would have to either pick a side or marginalize both of them. IOTL he sided with Cicero, which is probably what he would do ITTL. However, to sell that proposition to Caesar, he would have to guarantee that the new Pompey-Cicero coalition wouldn't turn against him. To do that, he may marry Sextus or Pompeia to allies of Caesar (perhaps betrothing Sextus to Octavia and Pompeia to Marc Antony or any of Caesar's other legates). There are honestly way too many factors at play here for me to speculate much further, since (in my view at least) the years 54 to 49 BCE are pretty much defined by Pompey's maneuvering against Caesar, and imho every event would be completely different. However, since both the 1st and 2nd triumvirates IOTL ended in civil war, I doubt that this hypothetical Caesar-Pompey-Cicero triumvirate would be any more stable.
Well, Pompey could have gone on his own military adventure while Crassus was invading Parthia, but he chose to remain in Rome. It seems to me that he had the opportunity to make quite a bit out of his situation, and simply didn’t.
I take issue with this. Pompey already had a credible military career including a war against Cilician pirates, the Third Mithridatic War, ending the Spartacus revolt, and the Sertorian War (among others). He probably saw more military conquests as a dead end since after the Third Mithridatic War he ended up being unable to pay his veterans anyways and needed the help of Crassus and Caesar to resolve his veterans' discontent. If anything, by remaining in Rome, Pompey was able to increase his power exponentially, and this is what eventually caused the Civil War of 49 BCE, rather than the senate doing anything to manipulate Pompey. The time Pompey spent in Rome during Caesar's absence was very productive from his POV. He was able to dislodge both Publius Clodius and Titus Milo from their demagogue-like statuses, recall Cicero from exile (who would then become a major ally of his in the senate), ally himself with numerous powerful senators including his eventual father-in-law Metellus Scipio, rig several consecutive consular elections, and even become elected as the sole consul in 51 BCE (which is effectively a dictatorship).
Pompey failed because he was simply a bad politician but his underlying calculus-that the triumvirate had only seen his political fortunes further deteriorate- was correct.
I'd say he rather failed simply because Caesar ended up being a better general, winning a decisive victory at Pharsalus. Our sources portray Pompey as a bit of a bumbling fool, but we should keep in mind that history is written by the victors, and the next 500 years of history are written by people who basically believed that Caesar was a god. If you read between the lines, Pompey was a deft politician that was simply in over his head, much like another competent politician/general from earlier in history: Cinna. Both Pompey and Cinna were excellent generals, popular and able politicians, but they lost because they weren't *quite* as good at being generals or *quite* as magnetic or charismatic as Sulla or Caesar. Sometimes history is decided by chance after all, and it's easy to say that either the victory of Sulla or Caesar was inevitable (hindsight is 20/20 after all), but we shouldn't write off Pompey as an idiot just because he was only the 2nd most powerful man in the world and lost to the 1st most powerful.