I think it all boils down to Rickard’s perception of Tywin Lannister, either not wanting him as a partner or expecting to be rejected outright. Tywin already had a reputation for ruthlessness with his actions towards the Tarbecks and the Reynes, as well as the punishment he wrought upon his father’s mistress after his father died. Coupled with once being Hand to Aerys, refusing to marry Cersei to anyone but Rhaegar (and then refusing to marry her off even after Rhaegar was off the market), perhaps Rickard doubted Tywin’s loyalty to him if swayed to his side and/or feared his ambition. Even with Tywin’s personal grievances against Aerys, Tywin was not one known for friendliness or general trustworthiness.

On the other side, Tywin may not have seen a northern alliance as a useful one, or a war against the Crown as a safe move. The North is far removed from the politics he’s involved in and is removed geographically. Then as we later see during the course of the war, Tywin does not involve himself until literally the last minute, indicating that he was simply waiting it out to see who the clear winner was before making his move.

Then there are Rickard’s own motives to take into account. He was developing a power bloc of nobles who were able and prepared to dethrone Aerys. Tywin, who was once Aerys’s close friend, would not be the safest pick. To approach him with a plan of war against Aerys would have been a risk Rickard was wise not to take. With Tywin once having been Aerys’s friend, he might have leaked such information immediately to earn himself royal favor. Robert was an especially good pick because he had no significant royalist influence, young, able-bodied, a skilled warrior, and the Lord of Storm’s End, having already come into his birthright (whereas Jaime was still merely an heir). His father, who had been a friend of Aerys’s, had died when he was young, he was close friends with Ned, and he looked to Jon Arryn as a foster father. Jon Arryn was receptive to and possibly even an architect of Rickard’s “southron ambitions”, he had both Ned and Robert under his care (while the two were becoming close and fast friends), and his loyalty to his foster sons was genuine.

Under such circumstances, when the choice is between Robert and Jaime, the winner is clear. Lannister gold means nothing if there is no loyalty attached, and I am certain that Rickard knew that Tywin’s loyalty would be hard bought and difficult to keep, whereas Robert could be easily swayed and kept to his side.

In any case, the unlikely match between Jaime and Lyanna would have disappeared into thin air upon Jaime’s induction into the Kingsguard, which would have been an entirely different mess.