Pretty good idea, it will definitely work. One suggestion, maybe put the aftermarket core after the factory one. I feel like since the factory one has coolant running through it, it might warm it back up.
I do Installs and flash tuning in South Florida
-2008 335i SGM on Terra 6mt
-1997 Jetta TDi 16v ABA Pt6266
-1997 M3 Pt6766