Biamping B&W 802 with Musical Fidelity and Bryston

The Bryston would be light years ahead of the Adcom, even though you're only talking about the lows. Are you certain the Toshiba is not the source of the harshness? Do you have an analog source available(TT or tuner) for comparison? Any triode based amp will give you a slightly relaxed top, but- a single-ended amp might not have the power to satisfy your SPL requirements with the 802's(even bi-amping). I don't know how loud you like your music(of course), so that's just speculation. I'm using a pair of modded Cary SLM-100's(98wpc into 6 ohms) to drive my Maggies(86db/1w/1m) and getting about 105db on peaks(without distress), crossed at 250hz(10th order) to my woofers(255wpc). You'll be crossed at a higher point(about 1 to 1.5 oct), but- I'd still wonder about your SPLs with an SET amp(your slope will be nowhere near as steep as mine). How many watts are you thinking about for your top? What kind of listening levels do you like?

I don't have analog source but I don't think it's Toshiba. Tweeters don’t sound too bad with well recorded music. Only bad recordings sound really harsh. Unfortunately a very significant part of my CD collection makes my eyes bleeding.
I can’t listen to music as loud as I wish thanks to the neighbors (I live in townhouse). I tried to bi amp my speakers with 30WPC Pass Aleph3 and 120 wpc MF and I could make my speakers sound loud enough. I think that I need at least 20WPC of tube power but I’m not sure.

Should you find that the bi-amping doesn't alleviate the problem, here's another option: If you open up your 802's: You should be able to easily find the tweeter section of the crossover. There should be two 3 ohm Vishay resistors in parallel with each other, and in series with the tweeter. If you take one out, that should reduce the output of your tweeter by somewhat more than 2db. Hopefully the assembler left enough lead showing for you to simply clip one, so it can be soldered back whenever need be. I'd see if I could find a higher-end CD player to audition before I went to that extreme though. Maybe even one with a tubed output/buffer section.

I could unsolder resistors, but I’d rather upgrade speakers if be-amping does not work. I already looked at Proac D25 but did not like its bass.It was a bit slow and not very tight. I prefer overall tonality and speed of b&W802. I’m thinking about Audio Physic.
I m planning to get a new CD player. I was considering Rega Apollo, but I see too many used on the market. It’s a little bit strange for a relatively new model. I’m using decent tube preamp so I’m not sure if tubed player or especially buffer would make a big difference.

I'd do the new player, or at least audition one, before any of the other options. I've never had a Rega, so I can't advise you there. I've got a BAT VK-D5, and love it(six NOS Siemens CCa's in the buffer). If your pre-amp is clean, it will convey whatever's fed it, good OR bad. A grainy/glaring/strident signal will be passed without mercy. It's possible you have some poorly engineered CDs, but- more likely that the player is the problem.