My first thought is that performance should be the criteria for changing the dremel not noise. If the issue really is noise, then replacement with a high speed spindle may not cure the problem.
I would change the dremel to a high speed spindle for performance reasons only and irrespective would encase the router in a cabinet to cut the noise. (or turn the radio up)
I agree re. upgrading the dremel mount and changing the linear guides.
The main thing that has to be removed is flex...ie complete rigidity is the aim.
Given you are using a dremel, then I presume your machine is small and therefore you'd need lightweight materials for rigidity. Aluminium is possibly the best material to use.
Good luck
Andy
Drat, imperfection has finally stopped working!!