You need more power. Separates is the only way to go if that's what you need.
Otherwise, you're stuck spending nearly 2k on a receiver that'll be obsolete in a few years.
Find a good 5 channel amp, used. I picked up an Anthem MCA 5 about 10 years or so ago for about 500 bucks.
It is an absolute beast. It's rated higher because of everyone using the "1 channel driven" numbers, but for this amp, "higher" is like 350 watts with a 4 ohm load like Def Techs are.
But it'll still do 170 watts X 5 channels at the same time. That'll run you out of the room.
You literally cannot stay in my room (which is a bit shorter than yours, but about the same width) until you make it clip. In fact, I've never heard it clip. Can't stand it any louder, and I LOVE it loud.
My speakers are Def Tech BP 2002's, Def Tech center and surrounds, and an M&K V75 sub. None of the Def Techs have any issue dealing with the power.
I used to have a B&K Ref 50 Series II preamp, but it developed issues and I bought a Pioneer VSX 1120 off Craigslist ($175 score!) a year or so ago to replace it, since it had preamp outs and HDMI that would pass 3D.
I highly recommend getting a separate amp. No reasonably-priced receiver is going to have an amp section like a good amp will, and you don't have to worry about the amp becoming obsolete. Hell, a good amp from 30 years ago is just as good as a new one today, sound-wise, other than the 5/7 channel convenience they come in now.
Get a good amp, then get the least expensive receiver you can find with preamp outs. Most of what you pay for in the more expensive receivers is the extra power. I say don't put all your eggs in one basket.
In fact, I'll never buy a receiver or preamp that costs 2k or more again. The B&K broke me of that. Just a few years later, and I needed HDMI and newer codecs. B&K advertised they could be upgraded and did not deliver. It sounded fantastic, though....but if you use the preamp outs with a good amp, they all should sound more or less the same, since they all mostly use the same surround processing chips anyway.
I just looked...there's an Anthem MCA 50 (about the same as what mine is, but newer) on Ebay for about 500 bucks+shipping. I'm sure there are other brands just as good.
Your main speakers are basically the next smaller size than mine, so they'll handle the power, no problem. Only issue is the smaller subs, but as far as overall loudness and how clear they are, a good amp should get you louder than you will want to listen to it.