Most powerful and popular EL34 tube amp?

Depends on SPL (in db's) you wish to obtain at the listening position. Both 89db speakers and 4ohms are do-able, assuming the 4ohm is their lowest impedance droop and you are driving the speakers a lease breaking levels. Do you have budget limitations? What kind of speakers are you going to drive?

I have a pair of your speakers in my closet. They are 92db and I had no problem driving them as loud as I needed in a 19.5x13.5x9 ft room with 40 wts.

The best sound I got out of these speakers using an amp with EL34's (SED Brand) was a Primaluna 1, a simple integrated unit which I'm sure sounds nothing like your BAT's. Probably warmer and less transparent/resolved. I found I liked the LSS's sound best, especially thru the mids and highs, using amps with 6550's or KT88's. In my room these speakers had a less than perfectly linear tone thru the mid/upper mid range and tubes with too much color (emphasis) at certain frequencies could sound a bit edgy. Maybe it was my room or ancillaries.

Power wise I tried 5 different tube amps from 40 to 160 wts and all had sufficient power to drive these speakers off 4 ohm taps, and in one case 2ohm taps, so finding an amp with power is not so much an issue I think as is getting one with proper tonality.

Sorry I couldn't be more specific or help. If I had to make a WAG I would suggest, in that price range, you consider a VTL or Manley amp using EL34's - they have a good reputation and a fairly modern sound (for tubes).