Yea, this might be just as good of an estimate as anything. It would definitely be the most straighforward.
My solution was just to sim a ton of them then find the ones with the same results. So if I simmed with 50 then print out results like 71,15,8,6 (for 0, 15 of 1 duplicate, 2 with 3 of the same etc for each trial.. Then just count the exact matches of each simmed pool size with our 1 sample. It would at least could remove variance on the testing side but there will always be the variance in the observed duplicates.
I'd have to think more about your method. It sorta makes sense but I'm not sure how good of an approximation it would be. However, writing a sim to test the 1 size estimated by the above amount would actually be quite easy. Hmm maybe when I have an hour or so free.
The curve fitting approach also seems correct to me but that is over my head. Maybe fresh out of college I'd have some clue..