As George said, room treatments will definitely help.
Rather than throwing the BAT out, why not have it modified?
As I look at the inside of the unit, a couple of things jump out at me. If you see those red cubes, BAT says those are "custom encapsulated oil-filled signal capacitors". To me they look like the Chinese paper in oils but I can't tell for sure. If they are,
that is why your BAT is the weak link. In my experience, those tend to sound rather dark and veiled. If any of those are Wima's, the same holds true.
Oh, I can't tell if they've bypassed the power supply caps (it sorta looks like they did) but I'd change out those Wima/Chinese PIO's for a decent cap.
If you want to
really invest some money, replace all of the power supply caps with Black Gates.....and yes, they are worth every penny IMO.

Next, it looks as if they are using pretty standard resistors in, not only, the signal path but also the B+ supply. There are some considerably better sounding resistors than the carbon films and carbon comps they've chosen.
Not sure if Bill at Response is taking on mods to products he isn't selling but he would be the guy I'd go to. The mods shouldn't run more than about $800 to $1k (pure guess) which leaves you ample for room treatments plus some money left over for some new music.
Those (type) mods, will gain you tons sonically. When you hear it, you won't believe it is the same preamp, honest.
It's
all about parts quality.