I believe that this might be the worst designed signature weakness in the LCG because it is missing 1 line of text. The missing line "Damage on this card may be healed as though it was on Vincet Lee". If you want it to be worse, make it healable as though it were on a different investigator, and you don't even get to give out on the mend while still healing others.
So why is not having this line of text so bad? Because it forces a character filling a niche role into running an even more limited subset of their cardpool or just praying that they don't draw their signature. If you do a search for all arkham cards that heal damage, you'll find there is quite a large selection of (mostly bad) cards. If you search for cards that can heal your ally? The list gets a lot smaller and a lot worse.
So to get it out of the way Hallowed Mirror exists and gives you Soothing Melody. For 1.5 melodies you can handle the problem (or 1 upgraded melody I suppose). This does assume you don't take damage in between drawing them, but I'll be nice on that front. From there, cards that could be used to heal your ally slot are:
Surgical Kit - Enjoy spending 3xp and 2 actions to resolve your weakness. Obviously this card isn't bad in Vincent, but it feels problematically mandatory if you want to be able to resolve bystander reliably.
Bizarre Diagnosis - Does it in one card. Admittedly its 3 actions worth of card because you're drawing a card, playing a card and dropping a clue. My read is that you basically never play this card except if you are trying to deal with Wounded Bystander.
First Aid - 3xp and 3 actions (after its in play) will resolve the weakness. This is notable because it can deal with the problem, but its really bad.
Inspiring Presence - Heals for 1. Worth noting but not capable of handling the problem on its own.
That's it. Those are the ways to heal your weakness. I'm not even discussing that thanks to encounter cards that hit all investigators (either in play or at a given location) for a small amount, you may draw this card and have it immediately pop in the same mythos phase. That bothers me less because sometimes weaknesses happen at the worst possible time. I get it. But of all the healing cards in the game, there should be more than 4 that "reasonably" resolve the damage healing character's signature weakness. I do want to acknowledge that Solemn Vow from another player can do work, but that feels almost like more of a problem than an upside. That's going to actively limit his deckbuilding even on the thing that he's most flexible for. And it didn't have to be that way. Both Foolishness and Rational Thought specifically avoid this problem with their effects, but it just seems to have been missed here.
And here's where my recommendation comes in and the real reason I think this is the worst weakness in the game. I think when you play Vincent, you ignore this card. You run the good cards that can deal with it, but you don't build your deck in bad ways otherwise. You don't run Bizarre Diagnosis just for this effect and you certainly don't go into First Aid (3). When you draw it, you either happen to have the limited number of good cards that deal with it, or you suck up the trauma. I firmly believe that the Doctor should just let the wounded patient die which is just a huge flavor fail, but I also think that its the best way to handle the situation by far. Again, if you have the upgraded Soothing Melody in hand, go for it. But otherwise, just accept that the Bystander is one more casualty of the mythos.