Jason, how convinced are you that your jaws are bell-mouthed? I ask because this is possible but excessive run out on a 3 jaw doesn't always mean it is being caused by the jaws. While bell-mouthing is not uncommon, wear in the scrolls or the jaw guides is a more probable reason.
The other thing is that run out on a 3 jaw is basically meaningless as a measurement unless it is done under proper conditions. A 3 jaw is a first operation chuck, meaning if you put some nominal stock in it and turn it then the work piece will be concentric with the spindle centerline unless you remove it; the accuracy of the chuck is not important. I suggest you mount a work piece and be sure that stick out is less than 2 times the diameter of the work piece. Then take a 0.010" depth of cut to get it round, then a 0.002-0.003" depth of cut and try to get a decent finish. Then, without disturbing the work piece or loosening the jaws, measure run out with a dial indicator. This is really the only way to measure the run out on a 3 jaw chuck. It should read zero run out. If you still have 0.03 - 0.04" run out then I would guess you have bigger problems than a bell-mouthed chuck.
If you do need to replace the chuck then I would think that a 4-3/8" chuck is a bit small for a 12" lathe and I would go for at least a 6" chuck. I would choose a 3 jaw if I were you. I own a 4 jaw scroll chuck and they are really good for mounting nominal square stock for turning; holds stuff solid. It also holds round stock just as well as a 3 jaw, possibly better, and it being a scroll chuck, it is just as fast and easy to use as a 3 jaw scroll chuck is. This 3-legged stool thing sounds like nonsense to me because even if the work is slightly off to begin with, it won't be once you take a cut in a first operation. The main reason I would go for a 3 jaw is because they are much more common and you're more likely to find a decent one.
You do not need a set-tru chuck for most first operation work, which is what most of us do. Set-tru chucks are useful if you have multiples of the same diameter to turn; in this case, they can be very accurate. They can be very useful for second operation work, where the work piece has already been turned and you are re-mounting it and need to restore accuracy. Most of us would use a 4 jaw independent or a collet for this but a set-tru is potentially just as accurate for this scenario. The misconception that you can zero a set-tru and have it turn everything that you put in it run dead nuts true is common but inaccurate. You can make a set-tru run more accurately than a non-set-tru chuck but for a first operation the accuracy of the chuck is a non-issue.
By the way, I own two set-tru chucks, a Yuasa 3 jaw and a Pratt Burnerd 6 jaw. I haven't set up the PB yet but the Yuasa is and it is a very fine chuck. It runs out about 0.0001" with the pin I used to adjust it BUT it runs out more for anything else I put in it for second operations work unless I take the time to readjust it for that specific work piece. For a first operation cut, it is as accurate as any other chuck I own.
Okay, back on track. I suggest you buy a decent 6 or 6.5" 3 jaw chuck. If you cannot find a chuck threaded for your spindle then buy a decent plain back chuck and mount it to a backplate that does fit your spindle. This will work better for you as opposed to buying that 4 jaw scroll chuck, at least in my opinion.
