Would a language borrow from another language a word with which it already has homophonous words in itself?

Yes, it would.

I’m not going to bother with examples other than grave (Germanic: tomb; French: serious).

It is a common perception that language change is driven by trying to avoid ambiguity. In fact, language has an astounding tolerance for ambiguity, because context usually takes care of it. Instances where words change in order to avoid ambiguity (eg French hui > aujourd’hui ) are actually pretty rare.