There are literally dozens of decent choices in low cost DACs From what I can tell, but I would take a look at the DAC module for your new integrated amp. Burmester doesn’t sell anything cheap, but there is something to the old saying buy once cry once
You would undoubtedly get the best synergy going that route, save a pair of cables And up your game while simplifying in the process. Otherwise, if you like warm, maybe something like a Lampizator? Again, they’re going to be people on this thread that will have better advice than me, so hopefully they’ll jump in. I can tell you from experience that once you start the upward progression, as you have, it’s going to bug you that you may have a component that’s a bottleneck.

