Maxx,
The precision (grouping) of a 1911 usually isn't the fault of the barrel bore condition. That's why I asked how you knew it was shot out. Usually it's the fit of the outside of the barrel into the frame that's 95% of the problem.
The rules of thumb I was taught were that barrel to slide and assembly pin lock-up is 65-70% of the improvement in group size. The barrel bushing is 15-20%. The slide to frame fit 5-10%. I've never heard of the barrel bore condition even coming into it, though in an extreme case of crown damage or a defective rifling job, or a lot of sand shot through the bore, it would. A truly shot out bore will have its throat gone, the dognut on the outside of the muzzle worn severely or some other factor that makes refitting impossible. But, in general, the .45 ACP runs at such low temperatures and pressures that you never see the kind of heat stress cracking of the throat than you will with a rifle, nor the kind of erosion you do in a magnum revolver forcing cone and throat until an extraordinary number of rounds have gone through it.
If you've never done this work before, I would go for the Kart EZ-fit, as you can usually get a good job with one without having to fit the slide and frame or the link lugs to the assembly pin or the link to the barrel, and still not have excessive barrel tip-up. It's also a good idea because Fred Kart will take the time on the phone to explain it to you if you get into a snag. It comes with a bushing, so you'll be taking care of the first two elements I mentioned and likely get 90% of any possible improvement without having to do anything else. You'll find fitting the bushing to the slide the hardest part. Before I had a lathe, I did it with shoe-shine sandpapering, a bushing wrench and valve lapping compound and patience. Slow going but not difficult.
If you plan on doing any other tuning on the gun, like smoothing the barrel tunnel in the slide or polishing the feed ramp or breaking the corners of the locking lugs, I recommend you do that first so you don't affect tolerances that affect the barrel fit
after already doing the barrel fitting. If you decide to try to learn how to tighten a slide to a frame, you also want to do that first.
When I first got my Goldcup in the late 70's, it would not hold a 5.5" group with lead bullets at 25 yards. After fitting it shot lead bullets into under 1" at 25 yards. Jacketed match bullets shot into under half that. Same original Colt barrel; just welded up and hand fit on the outside. And that bore was half a thousandth out of round when it shot those groups off the bench, so it's just not the primary issue in this gun.
I've put these up before, but it should give you some idea what to look forward to when you get all the i's dotted and t's crossed in the fitup.