Acuda touched on what I was about to say. I was in a similar situation a few years back. I had put together several .223 ARs and was looking to do something different. I also wanted something a little "harder hitting" than .223 and naturally looked to putting together an AR-10 platform. I ended up going trying out a 6.8 spc upper from bison armory and am glad I tried it out. You could look at that cartridge as a compromise between a .223 and .308...more ft lbs than a .223 but less than a .308. There are several articles that compare the ballistics of the 3 rounds. I have had no issues taking down KY whitetail with a 6.8 and it does very well out of a 16" barrel which is nice when I am operating out of a climbing stand. Having a short barrel in a stand is nice, and it's allowed me to comfortably take a few off handed shots comfortably when needed. Like Acuda mentioned, the deciding factor for many is that you can slap a 6.8 upper on your existing lower and be good to go. When you factor in the funds that didn't have to go towards a new lower, aftermarket trigger and stock, it saved me around $1k to spend on a nice optic. As far as ammo, factory rounds are plentiful with the 6.8 as well, and pretty affordable. Hornady Custom 6.8 SPC 120g SST were running under $1/rounds when I picked some up a few weeks ago from Target Sports.
I can't speak to using it in competitions but to speak to your "SHTF" scenario, my 6.8 upper is what stays on my primary AR. It hasn't replaced my .223, but has made it my back up caliber for the platform, as .223 ammo is much more economical to stockpile. I also like to keep my SHTF kit as consolidated as possible and it is much easier to throw an upper with a few mags in a pack rather than toting both a .223 and .308 complete rifle around.
To conclude, with your budget, I would utilize your existing lower(s) and put a 6.8 upper on one of them, but that is just me. I will mention for the past 5 years I have been trying to find a valid excuse to finally put an AR-10 together but the 6.8 is making it hard since it fills both the hunting and defense needs I have. I can't imagine trying to piece together a competition AR-10 with a good trigger and stock and still having the coin for a respectable scope for $1200 (or maybe you have an additional budget for an optic) but I would think it is absolutely possible to go the 6.8 route and be closer to your budget.
All of this is without knowing what you have into your (2) AR-15's part-wise. I think you will be happy either route, but just wanted to throw my PSA for the 6.8 spc into the ring.