So I've decided what I'm going to do with the dollar I won.
I'm basically going to attempt the recommendation offered in the comments by "sd_Baby-B" (who seems like an intelligent person full of good ideas so it's no surprise he/she doesn't comment here very often):
Write “Donald Was Wrong” as the title/caption. Every time he says something including the words “I think” simply look at him, raise your eyebrows, point to the framed dollar and say “yeah, but…” and shrug.
Then, when his hate for the dollar is at the highest maximum amount, bet him the same dollar that he can’t do the milk challenge. When he fails, frame your second dollar with the title/caption “Donald Was Wrong Again”.
This idea combines several of the suggestions I offered in my post into one glorious piece of Hannibal Lecter-esque brilliance.
My only concern with this plan is the risk of Donald still viewing the bet as a victory since he "almost won," and trying to make the dollar in to some sort of pseudo-trophy for himself. I'll be on guard against this.
Also since Donald might be too much of a pussy to ever do the Gallon Challenge even though he won't shut up about it, I'll also see if I can use the dollar to motivate him to write his "Why Hockey Sucks" essay he keeps talking about, which I will then post here for your enjoyment.
I have ordered a small frame for the dollar and it should arrive from Amazon in a couple of days. I'll keep you posted.