Ralph is the frame marked with the type ally steel it's made from? The tube diameters you mention and this coupled with the photos and your remark about burn through lead me to think this is a chromoly frame 4130 type steel which has excellent strength and welds fine, but typically wall thickness is really thin on bicycle frames to save weight. If this is the case burn through is likely when scabbing two tubes. Your plan to insert the stub into the larger tube is fine, I'd suggest slipping a snug fit length of tube or solid steel rod into the stub first to act as a weld backer it will help prevent burn through on the thin stub and if you do burn through a bit you've welded into the backer rod and the weld is still strong mechanically. If you drill a very small hole, 3/16" will do, in the sub before inserting the backer rod: and then weld a tack through the hole onto the rod to hold it securely in place, just in case you don't burn through and then have a rod rattling around in your frame...not cool!
Normally inserting the rod is used to strengthen the butt weld of two same size frame tubes, not only as a backer but to strengthen the tube in actual use. The plug weld or Rosette weld through the drilled holes would be larger and more numerous on both sides of the tube scab.
.090" wall thickness on ERW tube would be my choice unless .120" price isn't too prohibitive.
Rick C.