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

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  

The X12’s are 86dB sensitivity and rated at 4 ohms impedance, I am not surprised that the Brio is not doing it for you. NAD tends to be good with current thirsty speakers. Maybe Hegel, not sure. I would watch the used market and do your research, and focus on amps that do well at 4 ohms, and sound good, of course. 😀

Just taking a quick look out there, I used to have one of these:

driving my power hungry KEF LS50's, it does very well with 4 ohm speakers. The preamp leaves a little bit to be desired, but can be bypassed and use a separate preamp. I ran a Schiit Freya + into it and it sounded great. Lotsa room to move at that price. There are some others of this model out for sale, look around.

And not saying this is the best choice by any means, just one that I am personally familiar with.