I know I am in denial but part of me wants to think that giving her the money is a signal that I have totally "let go" or "given up" and that once I have given up the fight she might have second thoughts.
Might be easier for you if you think of it as "fulfilling your contract" or "paying your bills" instead of tying it to an emotional idea. It's money; not love.
Whether you pay now or later, it has no effect on whether she'll come back or not. And if the money makes a difference on whether she returns or leaves, well, that will be the best 20k you spend.
Don't you deserve someone who wants to be with you. not your checkbook? Don't you deserve that?