Square room is the worst type possible. Only way its worse is if the height was also 16 ft.
Is there any possibility, whatsoever, of moving the couch away from back wall? You will get multiple benefits, I can think of about 5 benefits off the top of my head.
I believe the subwoofer is a huge ingredient of the HT, but the problem is that apartments are not conducive. If this is true, Id start looking at tactile transducers, but the problem is that one of them can eat half your budget. However, even if it ate half your budget, its still something I would seriously consider. At the expense of your purchases, but that's just me.
There are a lot of nice products out there. It really comes down to personal preference with speakers. Check out one call for good KEF deals, they carry over 100 KEF products, and a lot of them with free shipping.
If you can do an upright bookshelf for center speaker, id look at KRK studio monitors. Perhaps Ascend 170 SEs for front 3, but problem is that's the other half of budget.
Too many ways to skin a cat.
Or you could do a pair of towers (if using transducer) and run them as fullrange. Most towers cant do full range, but at least you might get a bit more midbass when running LFE to transducer.
First off, tell me what you think of the transducer, look at Buttkicker and Earthquake first.
Cheers.