We also thought of taking out the four tiles that the screwholes
impinge on, and replacing with some kind of decorative tile of a
different style.

Seems to be a reasonable and cost effective solution. Some months ago there was
an article in This Old House magazine on decorative tile. Might be worth the
time to check out that and also ads in some of the magazines at your library
for companies that specialize in such items. That is, if you don't find what
you like at a local tile outlet. You've already done a Google search, right?
HTH

Joe