Hello Elberoth2,
I have heard many amps on the Eidolon over the years. Both tubed and solid-state. People say that they love power but I have found that IMO quality is more important than quantity. These speakers will give back to you exactly what you put into them.
My preference is for high quality tubes because I feel that it tells me more about an instruments sound, the tone, the timbre of texture, and the harmonics that make it unique from another instrument. Not to mention the space (venue) that the instruments were recorded in.
Of the tube amps I've heard personally on the Eidolon (I have since moved on to the Eidolon Diamond) either in my system or others, I prefer and picked the BAT 150se mono block amps. Other tube amps you may wish to consider are the Atmasphere amps that Rush (hi Rush) mentioned or the CAT amps you are already considering.
IMHO the BAT 150se has the best combination of virtues that both solid-state and tubes can bring to a system. They are a just a fantastic company to deal with and should be around for many years to come. In the 10 yrs they been in business they have become extremely successful and for good reason. They know what they are doing!
If you care to discuss this further please feel free to e-mail me privately. By the way, the speakers take a long time to break in. 500 hrs+ at loud volumes to really start showing their stuff.
All the best to you in your search,
Tom