I use a Cambridge CXC transport with my DAC, and it sounds just fine. As already mentioned, your existing Luxman CD player is a fine piece, so it might be better to do nothing.
CD transports
I recently started running my CD player ( Luxman D380) through my external DAC and have found a significant sonic improvement. I wonder if I would achieve further improvement in sound quality if I traded in the Luxman and bought a dedicated CD transport ( price range 3-4 K ).
Has anyone gone down that rabbit hole?