If that's what you want, I'd get an assembled AR upper from one of the better shops that offer them after shopping around. I'm not sure if by 'spending as little as I can' means 'I want it cheap' or 'I want to save some money even though I'm about to pay thousands'. Looks to me like at least two of those rifles have NightForce scopes on them, and you're aware that the price on these starts around $2,000 or more?
The stocks in the pictures include a standard M4 4-position, an A2 stock, and in the first and last pics what appears to be a CAA and an older MagPul PRS. They're all using some sort of free floating match tube forearm, a couple of them being CMMG.
The solution that Wendy mentions would mean you're going to need a specially throated match barrel with a lot of freebore for 80gr, and then a you're pretty much limited to shooting that specific manufacturer's 80gr cartridge exclusively. An 80gr 5.56mm round may or may not fit a standard AR magazine due to length, and as a result some match and varmint shooters using loads like these manually insert one round at a time through the ejection port and snap the bolt closed with the bolt release. Personally, I think anyone wanting to go down this path should stop screwing around with 5.56mm and just step up to an AR-10 in 7.62 which would make their lives and personal economy easier.
I normally load 223 with Sierra 77-gr bullets and have never seen any of the problems you mention.