[Eliot Ness]

I would first try going in from the top. Using some small long needle-nose pliers and hemostats you should be able to gut the thing and slip in the new cap and fish the leads out the bottom.

If you can't do it from the top I think if you make a thin/clean cut close to the bottom it will not look that bad when reassembled. You can find or fashion a small tube to slip both the top and bottom over and glue/epoxy them together.

I would insulate both leads and run them out the bottom and then connect the applicable one to ground under the chassis at a convenient location.

Worst case scenario would be to leave the old capacitor on the chassis and mount a new one underneath. That's the way an old repair shop would have handled it, and you could even stuff the new cap into an old axial electrolytic for a period correct looking repair. Just a couple of ideas for you to think over.