I think with some refactor that will do the trick!
On a side note, I have read a bit of the implementation of BilevelJuMP.jl and it seems their approach is very suitable for what I envision here. Generalizing it beyond two-player-Stackelberg games would work wonders for my case and would allow players to have constraints that depend on other players’ variables (completely out-of-scope for me right now, but very helpful for the community).