what replaced your Aerial 7B speakers?

Thanks for the explanation. Makes sense.

I have not done that of course. But, this is a 90 year old house that we renovated about 8 years ago.

Most of the wiring was original but was the old two prong type outlets. So when they added extra 3 prong outlets to the room they ran them from the panel and they do not share any lights etc. Just the new 3 prong outlets in this room. About 6. So I'm pretty close to a dedicated circuit.

When I'm listening the TV is off of course so the only thing plugged in and running in the room tends to be my laptop and a lamp or two.

So, not a dedicated circuit, but fairly close. Standard gauge wire and outlet of course.

I have owned the BAT and the bass can be a bit soft sounding.  I moved from Aerial 9s to LR5s (105 lb monitors) on 70 lb Sound Anchor stands and then added two, Aerial SW12 subs.  The combination of acoustic suspension (i.e., sealed box) speakers and the power and control of the SW12s have created as close to perfect bass as I have heard in my system.
Keep your 7Bs and add a pair of used SW12s on SA stands for your $4,500 and the sound will change your sonic world.

I honestly think there's more you don't like than just a bit more low-end. 
I don't want to put words in your mouth, but if you want a speaker with a 12 inch woofer or a facsimile thereof, fine. Get bigger speakers and hope they'll make you happy.  Take a long look at Egglestons, even used ones. 
They sound like music, have had nothing except very positive reviews by everyone, and go down to 19 Hz. If they don't do bass in your room, it's not the fault of your speakers. I had the Andra I and II, previously owned by the guy that sets up demo's all over the world for Wilson Audio ( a long story). He raved about them, also. Andre Previn (RIP) also had a pair. They have other models of course, a bit smaller, too. They are very well made, too- the grills are held on by invisible magnets. I can only recommend speakers I've owned. 

Egglestons are great speakers but they’re not going to offer anymore low end in a room that is 16ft deep than a pair of 7b’s. A 16’ deep room is going to go down to 35 Hz and that’s about it, which for most of us is fine because unless you’re listening to a pipe organ a full upright bass may hit 40 Hz. You could throw the largest Egglestons or whatever in there and it’s not going to change the physics of a 13’ x 16’ room. Also remember the OP is wanting to stay in around $4500. Maybe some room treatment or bass traps might be in order here.

EXCEPT DPM2340 was emphatic- he wants different speakers. Used Egglestons come up frequently and they are killer speakers, with one of the best tweeters ever manufactured, and a crossover designed by A.V. Schweikert. The driver integration is something you have to hear for yourself. They nail pipe organ music in my largely untreated living room.
I would agree the Aerials are more forgiving than some designs and are not something I would get rid of without careful thought, and I would try to audition everything I could. I have echo busters which are pretty good,
but I refuse to take all my pictures down and remove all the furniture.
I tried corner-busters and they looked like hell. One guy I know turns off his refrigerator and his air conditioner (this is Florida), has foam packing from boxes taped to the walls (real snazzy), and turns off the ceiling fan, too. Everything is suspended on little pieces of scrap wood. You can only listen for about 15 minutes before it gets so hot and humid in the room you can't stand it anymore. I have a different approach. I saved up (it took many years) and finally got to a nice level. My tastes are what they are. But when the time comes, I hate to cut corners. So maybe you spend 20% more than you wanted to originally- it might be worthwhile.
I would ALSO say that bass response is not that important, and it's really not. But then I would be hypocritical because I have a lot of pipe organ music.