xgboost or lightgbm to handle Binomial problems
I do not think this is possible with the main (as I can tell) xgboost library, because this essentially sounds like a regression on rates/proportions situation (a dependent variable bounded in [0,1]).

An easy way to have implemented this would have been to allow the Beta distribution (see how it is bounded) as an objective function in xgboost, but it is not an option at this point.