Adam Lane wrote:
Instead of taking it apart and re-applying plumber's putty, how about
if I just put some waterproof caulking around the flange? That would
be a look easier, but I'm not sure if it would be a good enough
solution?

If it were me, I'd take it apart and use plumber's putty. Without
that, it's won't seat and tighten up correctly, which is why you have
the gap. I guess you can try caulk, but it may be hard to make look
decent and may not last either.