Why would Rhaegar want hypothetical Visenya legitimised?
Well, the simple explanation is, why wouldn’t he? He went to an awful lot of trouble to make that kid. Providing for it would have been the easy part.
Granted, what seems obvious to me isn’t necessarily what would’ve made sense to Rhaegar. I can think of a few other explanations, though, none of which are mutually exclusive:
- Rhaegar’s interpretation of TPTWP prophecy did change over time. He thought he was the prince, and then something changed his mind. There’s no reason to assume he didn’t change it again. For all we know, his idea about “the song of ice and fire” got him thinking about the “ice” half of the song and made him think that TPTWP might not be Aegon, and moreover that this next kid may or may not have needed to be a literal prince in order to save the world. From his perspective, if you accept that the prophecy could be real - which, remember, is completely plausible in-universe; he was unarguably impolitic about the whole thing, but he’s not acting arbitrarily - legitimizing Lyanna’s line is a relatively easy action to take, to potentially critical ends.
- As we saw in TWOIAF, Rhaegar wasn’t uninterested in mundane politics, he just wasn’t prioritizing them. Maybe the plan was to have a child (or two) who was royal enough to be a potential consort for Aegon or Rhaenys without dooming them to his parents’ marriage or his grandchildren to the worst of the Targ inbreeding. I’m not saying the whole system isn’t gross, but from Rhaegar’s perspective so deep in Targaryen tradition, setting up half-sibling marriage or a branch of cousins down the line might not have looked too crazy. But that’s definitely not going to work if it’s legally a bastard line. Getting a little bit of paperwork (albeit temporarily fraudulent paperwork) in order could pay big dividends for his nearly-extinct dynasty.
- It’s possible (we don’t know, but it’s possible) that he wanted to convince Lyanna to leave with him and providing some legal security for her child/children and for herself (being the mother of a prince(ss) is way better than being the mother of a prince’s bastard) seemed like a good idea to one or both of them. That’s not to argue that anything which seemed like a good idea to Rhaegar was objectively a good idea, there’s some pretty good evidence to the contrary, just that it might have been his reasoning.
Any or all of those things sounds to me like a good enough reason to have one of his friends snag the royal seal for five minutes. Even if they’re long shots in terms of future enforcement, it’s better than no shot.