My plan is to buy a 700 (a cheapish one if possible) in .223 and put it into an AI chassis and get a custom barrel. And maybe later down the line when I can get some money together I might replace the action and trigger. But for now its just custom barrel and the AI Chassis.
This plan doesn't make that much sense. Realize that when you "replace the action" you are essentially getting rid of the gun. Also, why would you buy a new rifle at a gun store only to chop it up? Why do you need to constrain your purchase to your local dealer? There are many more options available that would provide you more satisfaction.
Let me propose to you a few more options:
1. Go to your local pawn shop or gunbroker.com and look for any used Rem 700. Use this mainly for a donor action. Keep in mind that many of these will be of the hunting variety and may be a long action with a .308 boltface, which is fine if you plan on shooting 30-06 or 270. But you probably want something with a short action. You may luck out and even find one with a 223 boltface, but don't hold your breath. Then, buy yourself a match grade barrel and your AI chassis. Send to a gunsmith you'll have a semi-custom rifle. This route will probably run you around $2K.
2. Get yourself a Rem 700 5R in your desired caliber (223, 308, or 300 win mag). Milehighshooting sells these. Milehigh also sells AI chassis. They would probably send you the 5R already dropped in the chassis and mounted with a scope if you asked them. In my view, this setup is a great compromise between a factory setup and a full blown custom and is the route I recommend. This route will probably run you about $2k also, but it will be much quicker than option 1.
3. Get a Rem 700 AAC-SD in 308. It has a 1/10 twist, which is well suited for heavy bullets, and already comes with a threaded muzzle (BTW, so does the 20" .308 5R). Upgrade as you get funds. Start with the trigger, then reloading gear. This should be your second option. It's less expensive than options 1 and 2. You can get good results with a decent trigger upgrade and handloaded ammo, but not as good as options 1 and 2.
4. Save your money for now and get what you want. Ammo is a real bitch to find now because an old lady is proposing a gun law in the US Senate that has absolutely no chance of becoming of becoming law and no good business enterprise lets a crisis this juicy go to waste. So, now is a perfect time to sit back and have your custom gun get built. Your new rifle will probably be finished when the bubble pops a few months from now. I recommend GAP's Rock rifle with the Badger M5 upgrade. This will be about $3.3K
5. Get one of the rifles listed above. It gets you in the game now and out shooting, which is most important, provided you can find ammo. None of those options you listed are anything I would buy, but it's your money; do what makes you happy.
Hope this helps.