I wonder if it's possible to derive a formula to determine how much money you'd need to raise it by what percent...
I doubt you could come up with anything too precise, as the number of unknown variables is just too big (how many people out there have large holdings of BTC, for instance, and at what price point would they be persuaded to part with theirs... there's just no way to know). It would also depend highly on how obvious the move is. Presumably, if your intention is to raise the market cap, you'd want to make that intention fully known and then follow through on it.
And again, raise it to how much, and starting with how much?
Too many unknowns.