I didn’t notice that you are working with sparse QR. That is even more tricky. See: Differences in A \ b for sparse and nonsparse rank-deficient A - #5 by stevengj … certainly, you can multiply by a subset of the columns of Q efficiently as explained in that post, but I’m less sure about what is feasible to do with JuMP.
(This may be one of the cases where JuMP is too limiting and you need to call the lower-level optimizers directly.)