Crossrail 3 would then takeover the Wimbledon slow lines and serve
Vauxhall, Battersea (new platforms),

What's wrong with the old ones? Or are you saying that there should be
new platforms on the Victoria to Brixton line?

The problem with Queenstown Road Battersea is that it's only on the
Windsor Lines. The proposal is basically a SW (Main) Slow Lines
takeover.

Is there room though? There are only two (used) Windsor Line platforms and
the faster trains don't call.

The faster trains not calling there reflects the level of patronage.
As you implied, there is of course a 3rd platform, albeit on the
Windsor Reversible Line (ie the relatively useless one). As for
putting platforms on the Main Slow Lines, there are two big problems -
the constraints imposed by the Atlantic Line viaduct overhead at the
Up end and the BML itself being elevated (Queenstown Rd viaduct) at
the Down end.