Welcome to TS!
This is a cool idea, and it sounds like your home is a really great setup for this type of thing. Some thoughts:
Heathen is 100% correct that you should devote most of your budget to your mics. They will make much more of a difference than a fancy recorder. So, sorry that means no Sound Devices in your budget.
For the type of music you are recording, you should never need anything more than one stereo pair of mics. So for a recorder, I would recommend a
Tascam DR-100 mkIII. You've got 2 phantom-powered mic inputs, and a dual record feature so you have a second set of safety tracks recording at a lower level in case you set your gain is too high. (This is
always the way to go for acoustic music. A limiter should only be used as a last resort.) You could even spend a bit less on a DR-70D if you think you need 4 mic inputs, but the preamps and DAC in the DR-100 are better. That's probably the best bet for a one-box / small / cheap / high quality portable recorder on the market right now.
For mics: While omnis are my favorite to use in a concert hall, they will not be a good choice here. While you have a large area, your ceiling is too low which will cause too many early reflections in your recording with omnis. My main floor has 11' ceilings, and in my piano studio it peaks to 14' and even there I won't use my omnis.
You want a pair of small diaphragm cardioids. The
Superlux S502 punches way above its $200 price, but then you're locked into the ORTF array. Better off to get a separate pair of mics for flexibility, unless you're often recording in great-sounding churches and concerts halls that have acoustics ORTF works well with.
There are plenty of super-cheap mics out there, but most of them are going to be limiting for acoustic recording. So within your budget:
Jon at Naiant does excellent work, and you can get a pair of his
X-Rs for around $200.
Next up would be the
Line Audio CM3. Click the Team Line Audio link in my signature to learn more about them. About $600 / pr at current exchange from NoHypeAudio. I use these heavily and they work great in a variety of situations, including a lot of lousy-sounding school auditoriums.
The last great mics in your budget I'd recommend are the
AKG Blue Line series at $1000 / pair. These have the benefit of separate bodies and capsules if you want to change patterns later, it will only cost $200 per capsule.
To get better mics than that, you're moving into the true top-tier - Sennheiser MKH, Schoeps, Neumann, Gefell, DPA, Josephson, etc. Just one of those mics is going to take up your entire budget (or more).
Finally, get a custom set of mic cables made one of our cable makers here -
GAKables or
Darktrain. Both do excellent work. Jon at Naiant also makes great custom cables.