Who do you think would be most likely for Aerys to offer Rhaegara’s hand
to of the great houses, if not Robert? Would Jaime have been a viable
choice?

To answer that question, we have to identify what Aerys would want in a bridegroom for his only daughter: 1) that he has Valyrian heritage, 2) that he’d be from a powerful family so that the crown can build a power bloc to counterbalance Steffon’s SA bloc, and 3) that he would not be someone that could threaten Aerys’ own throne.

The first requirement restricts us to only two Westerosi families with the exclusion of the Baratheons: the Velaryons and the Martells. A Velaryon would be the most likely candidate since he’d meet all three of Aerys’ requirements quite nicely; of impeccable Valyrian heritage and looks, the House that proudly boasts of being The Old, the True, the Brave would make for a more traditional nuptial choice for the Targaryens, and to a king who attached much weight to Valyrian blood and leaned towards traditional Targaryen marriage practices. Certainly, the presence of Lord Lucerys Velaryon, noted to be one of Aerys’ cronies IOTL, on the small council would not hurt the chances of getting one of the scions of his House married to the princess since Aerys was known to lavish honors on his lackeys, and since the king would be able to trust that his loyal supporter would not try to use the princess’ claim to usurp his throne. In terms of power, House Velaryon is a significant sea power with one of the strongest fleets on the continent that provide the majority of the crown’s royal fleet, though it’s unknown how many foot-soldiers they could field, or how much wealth they retained by the reign of Aerys II. I’d say they were still pleasantly wealthy due to trade with the Free Cities, even if it was in no way close to the heyday of the Sea Snake, and due to Aerys’ own tendency to shower extravagant gifts and lands on those who pleased him which certainly only added to the wealth of the House of his loyal courtier. It’s a solid match, even if I do not think it’s the most advantageous in terms of creating a power bloc against Steffon’s own (the Lannisters or some Reacher family like the Hightowers would serve him better in that regard) but it’s one that would appeal to Aerys.

As for the other Valyrian-blooded House, I’d say that it’s highly unlikely that Aerys would arrange a match with House Martell. He might have married his heir to Elia IOTL, but I believe that was an attempt on his part to curb Rhaegar’s power by limiting the amount of support he could garner from his wife’s family to then use to overthrow his father. That’s not to say that the Martells have no advantage to offer the crown: they are a prestigious Great House that has intermarried with the Targayens twice before making them a more traditional match, and they’d offer a strategic advantage in forming a nice choke point around the Stormlands, alongside the Crownlands and the loyalist Reach if it comes down to war. However, Dorne is relatively weaker than most of the other Great Houses, and suffers a cultural divide from the rest of Westeros that makes the Dornish the target of racism and derision. This alliance would not offer Aerys the military advantage he seeks or help him solidify his political position, or be particularly accepted by the wider (racist) Westerosi society who would resent having a Dornishman as prince consort, and that’s without taking into account Aerys’ own deep racism and prejudice against the Dornish. Oberyn was also already notorious for allegedly poisoning Edgar Yronwood by 273274, and Aerys might have worried that having the Red Viper in his court might not have bode well for him once Oberyn got a child on Rhaegara.

Those two Houses are the main candidates that fit the traditional Targaryen marrying practices that Aerys seemed intent on returning to, though it’s worth noting that there are actually other families with Valyrian blood in Westeros like the Celtigars who are rumored, in the current generation at least, to be quite wealthy but we have no info on the extent of their militarynaval power, or why they were never considered for unions with their Targaryen overlords as opposed to the other Valyrian-blooded House, the Velaryons. The Celtigars generally seemed a bit shut out of the Valyrian power bloc with only three Celtigars holding high offices for their Targaryen kings (to our knowledge anyway. We hardly have an extensive list of every person who served the Targaryens), and no Celtigar maiden ever marrying into the Targaryens (or vice versa). They did not seem as involved with the royal House which, compared to the Velaryons, seems a bit odd, especially since the Celtigars were allied with the Targaryens even before the conquest.

But back to questions of suitors. Any other potential matches for Rhaegara would have to make away with the Valyrian blood requirement (because we have no knowledge of any extant Targaryen lines other than that of Princess Rhaelle’s and Princess Daenerys’.) and that’s where Jaime Lannister falls. The lack of Valyrian blood is a strike against him from the get go, but how well does he fit the other two conditions? In terms of power, House Lannister tops the list of advantageous marriages. Their wealth and power is undeniable, and a marriage alliance with Casterly Rock would form a nice power bloc alongside the loyalist Reach to counterbalance the SA one. Strategically, the alliance can be used to harry the borders of the Riverlands and threaten the Stormlands from three directions if it comes to outright war. It’s an excellent match, if it was not for Aerys’ mounting suspicion and resentment of Tywin. Even with his relationship with Steffon potentially on the rocks, I see no reason for him to suddenly trust the Hand whose overreaching power threatened Aerys’ own. An alliance with Tywin might curtail Steffon, but who would curtail Tywin? What guarantee would Aerys have that Tywin would not seek to depose him once Jaime and Rhaegara have a son? That would be getting out of the frying pan and right into the fire as far as Aerys was concerned.

And that’s even assuming that Tywin would suggest the match. It seems like a no-brainer but Tywin was cautious when it came to his children’s hands, intent on holding off on betrothals till he could ensure the most fruitful match. The hand of a royal princess is most definitely the most valuable marital prize in the land, and Tywin could do a lot worse than having a Targrayen princess as the chatelaine of the next generation of Lannisters and adding more prestige and power to his House, even if she did not end up inheriting the throne. But Tywin’s main purpose in seeking a royal match was to have a future grandson on the throne, and that might have not seemed guaranteed in light of how Aerys and Rhaella were regularly conceiving and even produced three living princes, even if the infants still went on to die only a short time later. If Tywin betrothed Jaime to Rhaegara, only for Aerys and Rhaella to have a healthy male heir, he’d have missed on the chance of having a royal grandson. More importantly, Jaime was only 10 when Prince Viserys was born and so any attempted matches would have had to happen before that. Offering a 9-year-old as a viable match to the older princess simply was not tenable: Rhaegara would have had to wait at least few more years before she could start having children which was not feasible for a dynasty on the verge of dying out, or a paranoid king desperate for a leverage against a competing line. And once Viserys was born, Tywin’s efforts were better utilized in trying to secure a match between Cersei and the little prince.