You can reuse your chuck , there should be nothing wrong with it.

The shop should be able to get the end off without damaging too much but you need someone with a JIG that can weld the end back on straight. there should be a driveline shop somewhere in your area , or find a shop that builds race cars , they will have the jig.

The housing may end up a tad shorter, the thickness of your cut? No biggie, that is lost in the noise.

I think you may have cut too much off though if you cut if off beyond the part of the housing that holds the race?picture I can't see ? That part of the housing is messed up if the race is stuck in it and should be replaced with a fresh end, I'd not be too confident in it holding the race from spinning when reassembled.

You might be better off just starting with another housing?