What Power Amp matches with the Rega Elex-R ?

I would like to improve the sound quality a bit, because the upper mids appear a bit too pronounced and I could use more specific placement of instruments in the audio spectrum, IMO.

I think you should start with some room treatments on the walls to absorb standing waves. This is the first step in trying to tame upper mids and highs. Think about it, bare walls create a perfect surface for upper frequency sound waves to bounce around the room.
Try the "clap test"...clap your hands and listen for any echo in the room.

For better imaging of instruments, first try pulling your speakers forward, away from the front wall. Also try varying degrees of toe-in.
That’s a good sized room, maybe move the speakers farther apart.

very nice set up. I dont know your model of Tannoy but my experience with Tannoys is that they dont necessarily respond the way you might think to more power like other speakers. They can change for the worse at times.

moreover, an issue with upper mids isnt usually solved with more power as far as I know. Usually a different pre, a different dac or even cables could impact upper mids - and most importantly placement. 

just for fun could you try the system in a different, smaller room with thicker carpet just to see if you have a room resonance problem? also what is your source? if digital then look at the file types and experiment with others - even plug an ipad into the rcas just to see if the problem changes. try a different dac. sometimes when you put together a good system like this you find out the upstream needs work because you hear the problems more clearly. 

enjoy the system and keep us posted.