A lot of transformers have a bunch of extra leads. Which is fine. You can cut them off if you want, but if you decide to use the transformer for something else down the road, you're pretty hosed. Typically, you'd cut the leads short, but accessible, and tuck them away somewhere. Is there any reason you couldn't remove the end bells, and (assuming there's space) tuck them up in there (also assuming they're well insulated)?
You’re asking for trouble by putting them inside the end bells. Heat, being pinched, shorting out, etc.. I just put a piece of shrink tube on the end and wrap it up in a small zip tie. Inevitably you will use the transformer in another project and kick yourself for cutting the lead short.
I took the transformer (Hammond 372X, FWIW) apart this morning, and there's quite a bit of room in there. Nearly all of it where the wires exit the end bells. I measured the wires out so I could fold them over a few times within the length of the cavity, heat shrinked them, and put the end bells back on over the top. The first time, I had a few too many folds, and it was too tight for comfort (probably could have forced it closed if I tried), so I cut it all back apart, removed a fold from each extra wire, and heat shrinked them again. Ditto the other side, but with fewer wires and lessons learned. Between the wire insulation and heat shrink, I don't think there's any remote risk of shorting. There's enough room to not be worried about any pinching worse than what I did when I folded the wires over prior to heat shrinking. Unless the transformers are actually reliant on some sort of air circulation through the wire holes through the end bells (which seems extremely unlikely), I'm not that worried about heat. That said, I think heat is likely the biggest risk. I'll keep an eye on it, and maybe pull the transformer apart again after several hours of operation to make sure there are no issues, but if those are the only concerns anyone can come up with, I think it'll work out well.
I wouldn't worry about heat. On many vintage transformers, the only part of the wire insulation that was OK was what was inside the end bells. Outside was petrified - brittle, crumbling... inside had original colors, was flexible.
I've done the same thing on the last couple of builds, double heat shrink and stuff them inside the end bell leaving enough length for future repurpose.