Having had numerous electrostatic speakers, I can only tell you that Solid State is not really the right choice. The impedance swings are too wide and often the speakers have too high impedance in the mids, which leads most SS to sound thin through that region on the speakers.
To avoid back EMF issues you need a good tube amp without negative feedback. If you look at something like VAC, I would go back in time a bit and try to find a pair of the PHI 70 monoblocks. These will deliver plenty of power to your CLXs and sound extremely good. Nevermind the 70 watt rating there is plenty of head room and they will deliver the voltage your speakers need through the higher impedance midrange. Other big push/pull triode or triode wired amps should be considered like the CAT JL2 signature (it should be stable). The Coincident 211 based push pull monos should also be on the audition list. You shouldn't discount big SET amps either from the likes of NAT (SE2 or Transmitter) or KR Audio (Kronzilla DX monos will work beautifully with the CLX), which would be my own personal preference over the push/pull designs.
Interesting, my experience is exactly opposite. I find tube amps struggle with the low impedance ESL's present at the top end, making them sound more edgy than airy, and do not control panel modes in the bass, making them sound more boomy/less tight than a good SS amp. And of course "good" means it must handle the low load.
An amp with little feedback and high output impedance by definition will not handle back-EMF as well as one with very low output impedance.
I love tube amps in the middle, and can live with their high end given they tend to roll off when presented with a low HF load, but unless it's a very large amp tube bass I find lacking. Overall my preference is for a great SS amp on a great ESL.