As I noted in my comment, you don’t need two factorizations, you only need one. The QR factorization of A (which you use to solve the least-squares problem) automatically gives you the LQ factorization of AT (which you can use to solve the overdetermined problem).
Oops, forgot to strike out that part of the comment! Yes, you only need one factorization but still require two backsolves (as I noted after the edit); but yes, having that utility would be quite nice generally 
. This solution would still roughly double the current runtime, though, and I’m not sure the gains in precision (especially for a low-moderate precision method such as ADMM) would be worth it.