"If I take a smaller frame, and remove the bottom tube, and cut the top tube in half,
And then add 3 or 4 inches to the top tube with a slip-fit tube that fits tightly around the original top tube ends,"
I sometimes comes as a surprise to people that there are not nice little tubes available all designed to slip fit to each other. But there aren't Tubes are defined by outside diameters and wall thickness, this means that there are rarely good ferule like fits. They can be reamed, but you probably just blew your budget in that word. Pipes are defined by internal diameter, and you would not need to use 4130 for this kind of job, but a 1.125 pipe will not fit over a 1.125 tube, without reaming. Of course if you treaded this as a demountable, you could slit the tube, braze on fittings and crank the thing together with screws, and you might even get away without reaming. Your splice can be glued on, by the way. You can glue tube butt joints, but you can glue a sleeve.
"And then put in a new bottom tube from crank to headtube,"
Essentially this tube can be a cable, so at some level of theory you cut the tube out and replace it with a lot less than a tube. Since a lot of your problems with jigging and alignment come from taking this tube out, I would consider something like a demount system in this area designed so as to leave the original tubes in place, look good, and have some function. Putting in a new downtube is non-trivial.
"It would only be 4 welds. The top 2 welds would be simple to do well; If I use an elliptical tube for the bottom, the whole mess would be even stronger than the original because the welds from bottom tube to bottom bracket, and bottom tube to headpiece, would be longer. I would MIG weld it."
You were good right up until the MIG weld comment. I have done a lot of work, perhaps as much as anyone, on non-standard welding. Both because I wanted to figure it out so folks could build cheaper, and also for the old roadside touring welding thing. MIG is possible, but you should look up the many pages of stuff largely directing oneself away from it. Any system that doesn't allow one to control the puddle and the filler independant of one another, in real time (not just some presets), is asking for trouble. It should be said that almost anything you can imagine where welding is concerned is either being done, or was done for some material economy reason during one of the world wars, or something even kludgier was made to work... But getting from that to your first build is a whole other thing.