What sort of upgrade from Rega Brio to go with my Dynaudio's X12?

^^^^^^^^
Sounds to me like you just want to spend $$$...
 A Tube/SS hybrid will give a slight upgrade in sound but a dramatic improvement will only be accomplished using those speakers with a full tube or pure Class A SS amplifier...They are decent enough speakers BUT thay have a pronounced mid bass hump & slightly recessed midrange..
IMO you will be better served keeping the Rega for now & upgrading the speakers..

interesting comment. that was my first idea, in fact. I thought of getting Monitor Audio's Silver 100 7G people talk wonders about, then was told my speakers were too good for the Brio and that I had not seen what they were able to do with more power, thus, this post and, now, A-S801 more power, very well reviewed amp, etc...

I believe you are the right path upgrading your amplifier for the X12s. I have heard and truly like them. At a budget of 'up to 2K' you can do much better than the Yammy 801 in my opinion. 

What sources are you using?

Cool. Was using Bluesound Node and now experimenting with the Wiim Pro Plus. 

Consider the Heaven 11 Billie integrated amplifier mentioned above.  Not pushing it, just very satisfied with it.

Complete with DAC and phono.  

Rich