Just wanted to chip in with my personal experience. My living room is >5000 cu ft and I started off with a Dayton sub-1200. I think it is a fantastic sub for the price. It actually provided a decent amount of bass in my living room. A lot depends on how loud you tend to view movies. The Dayton would likely have issues at reference levels in that large a space, but for someone who didn't have much of a sub before (A logitech z-5500 system), the Dayton performed admirably.
A week back, I picked up the SVS NSD-PB12 for $500 and it is absolutely fantastic in my living room! I know everyone here will tell you that you need 15 inch subs and a pair of them at that, but a decent single sub can do an adequate job in your setup. Sure, double subs would be nicer and lead to smoother bass response, but it sounds like you are on a budget and 2 subs is unlikely to be entertained.
If you are getting a single sub though, you need to be flexible with sub location, and you need to put in the time and effort to take measurements with REW (or manually using Excel and an SPL meter - my phone mic worked fine) to ensure that you find an optimal location for your sub. If you are lucky, the corner you have allocated for the sub will end up working well. But you will be amazed at how minor changes in position can have HUGE impacts in terms of the bass response curves at your seating position. Moreover, since you will have multiple seating positions, you will likely have nulls at specific locations so you'll just have to optimize for a sweetspot, or optimize by taking an average measurement across seating positions.
In my case, one of the room corners worked fairly well. This is the response I measured with a single SVS sub at the corner in my 5000+ cu ft living room with some minor EQuing on my HTPC with Equalizer APO:
Seating position averaged response:
2-3 subs would make for a much smoother bass response across the entire room but you still need to put in a lot of work to optimize and tweak all the subs locations, phase settings, delays and gain levels to optimize the overall response.
If you do not take the time to find a good location though, you could end up with some horrible sub response resulting in a very underwhelming experience (2nd, 3rd and 4th plot...this is for my Dayton sub-1200)