Yes you nailed it, this is exactly the problem. The right thing to do would be to explain to ForwardDiff how to differentiate this function.
Also related, from 5 years ago, same issue : Differentiation of incomplete Beta function with some code for the gradients apparently. Also I have seen the same problem when working for Studient copulas, or when fitting multivariate students…
Opened an issue there to get some guidance, if we want to fix this once and for all.
Edit: aaaad did it myself in this PR, which should solve your issue.