The 220 will require additional work to the bolt face if you're going to switch it to .223.
As for the price on the barrels, I just checked Riflebarrels.com which is Lilja's website, and I believe you're going to be looking at $295 for a chromoly barrel. Then you're going to be paying a gunsmith to fit it, and finish it (it'll have to be blued, as they come "in the white" from Lilja". Lilja does not do any of the gunsmithing/installation. They just ship you the barrel you want in whatever contour and caliber you ask for then a Gunsmith does the rest. The Gunsmith work is going to be a minimum of $200 and that's really a low ball. Considering it'll have to be blued, and the bolt face work, you're probably looking at more like $400+, but I'm just shooting from the hip here.
Within your budget, I'd either go with something that is pre-packaged, or hold out a little more cash and go full out custom.
For just a shade more (about $250) than what you'll be paying for the 220, you cut get a custom Stiller action which will be accurized from the factory, and is going to be one of the best that your money can buy. Then add to that a barrel, fitting etc. The Stiller isn't going to come with a stock and trigger, but it's a Remington 700 pattern action, so the options are plentiful. Obviously this is going to up your price about $500 or so, but when it's done you'll pretty much have a competition grade rig. Just a thought.
Otherwise, I stand by my previous statement. Go with something that is already built to your specs. Unless you're starting out with at $100 surplus gun, going the custom route is rarely going to come out cheaper.