OK, I'm overcomplicating things but here goes with a very long-winded description:
On your Les Paul, the mounting ring is angled, the screw-heads sit against the (angled) top of the ring and the springs push against the (angled) bottom of the ring. There's nothing pushing in any other direction, so the pickup happily sits parallel with the angle of the ring - and therefore parallel with the strings.
On a V, the (angled) mounting ring is sitting on top of a (horizontal) scratchplate - so, as before, the screw-headss sit against the (angled) top of the ring. BUT, the
springs push against the (
NON-angled) bottom of the
scratchplate. So, if the springs are under a lot of pressure, they're trying to push themselves into a straight line - which pushes the pickup back parallel with the scratchplate, rather than parallel with the strings. Which is the problem you had in the first place.
As I said before, you can get round this by cutting the scratchplate so the springs are no longer touching it. Then everything depends only on the angle of the mounting rings. Just like the LP.
OR, you could use very short springs so they don't get compressed much - in that case, the screw-heads sitting on the angled pickup rings will give you the angle you want, and the springs will just bend/flex a little and won't have enough strength to change the pickup angle. Only slight problem then is, the pickups won't be sitting very solidly, they'll wobble a bit if you rest your hand on them.
I said it'd be long-winded.

(I had the exact opposite problem with my own V. It has a late-70s-style neck which
isn't tilted back - it's parallel with the body. So I
wanted the pickups to sit parallel with the scratchplate - but they'd used angled pickup rings. I replaced them with non-angled ones.)