I would urge you to consider that for good Subwoofer performance in that room, you will need 2-3 15" subs that can handle Very Large or Extreme rooms... I wouldn't recommend anything less than that.
The DIY angle, if you are so inclined could serve you very well here. Cost for performance is greatly in your favor with this, and you could easily build 3 18" behemoths for a mere pittance. You need space and tools, though, and some measure of skill if you want them to look good. If your time is valuable to you, and you lack the space and time...
...I would urge you to consider any of the usual suspects we recommend to our friends engaging in this pursuit.

Hsu, SVS, Monolith by Monoprice, Outlaw, PSA, Rythmik... and for a slightly higher ticket: JTR, Seaton, Deep Sea, and Funk. To be honest, the value leader right now seems to be Hsu, and 3
VTF 15H mk2s would be a great way to handle the bass needs of your room within your budget. (That link is for 2 ea, but if you call, they usually will give a discount on adding a third as well.) If you want to save a little bit, their VTF 3 is almost the same product with slightly different size and extension and will cost a little less.
SVS gets expensive, but the PB3000 or PB/PC4000 would be good options. You've already eyeballed the Monolith 15s. I use the Outlaw X-13s and can vouch for their quality.
At the end of the day, most of the Subs from these companies tend to be good products and will perform well.
Your room is going to be challenging, somewhat, just because of the sheer size. The fact you have Cathedral ceilings though is nice and will help your acoustics out by breaking up the parallel surfaces.
Please look at the companies I mentioned and ask some questions! Especially if you think you'd be interested in the DIY approach. There's a lot of great cat's here with varied experience, and we'll gladly help you out if we can!
Cheers!