I'm probably not doing this right, but I'm taking my shot.
To start out I made a round tube of 2/3 oz leather for the lower few inches of the scabbard to go under the worst part of the damage--that turned out to be too bulky, and while I was figuring out my insert was too fat the stitching pretty much unzipped all the way up the scabbard. I ended up cutting the insert down to a half-round reinforcement that fit under the seam. After making the insert I brushed on some glue and slid it in so it pressed against the underside of the seam. Then, while the glue was still wet, I glued the full length of the seam back together, edge-to-edge. I used rubber bands over wax paper to hold it all together while it cured and inserted the bayonet to make sure it didn't get squeezed too small. I then set everything aside while waiting for the postman.
Following up the on the lead about red rot (thanks St. George), I did a little research and found a product that supposedly retards the disintegration--the one I got is called Cellugel, but there are others with similar names. This stuff is apparently intended to restore the leather bindings of antique books. Anyway, it soon arrived, and I put it to work. The product is a gelatinous clear goop that reminded me of a jellyfish--it seemed to have a little more body than edible jelly. The instructions are sparse, and, I suspect, geared toward repair of thin leather book bindings. They suggest an initial coating and, after it dries, a second application. The idea is that the stuff soaks into and binds the leather fibers together.
I was concerned that with the jelly-like consistency it was not soaking into the leather as deeply as it should. The label says it can be thinned with 100% isopropyl but doesn't say why you would want to do that. I only had 91% isopropyl and 95% ethyl--I went with the isopropyl and will save the Everclear for a different application. I was not precise with my measuring, but a mix of about 50-50 was thinner, but not so runny that it would drip off without soaking in. I ended up using one thick and three thinned applications, plus I slopped some on the bayonet and inserted into the scabbard to try to get some on the inside surfaces.
All-in-all I'm pleased with the result--still have some cleanup of extra glue to work on, need to decide on some kind of finish, and also need to figure out a finial; I'll probably just leave the stitches as-is, because I'm not really a leather-guy and would most likely just mess them up. But the scabbard seems much more solid and like it will stay together, at least for awhile longer.