Omega Constellation 1333 - How much does would it generally cost to swap movements?

Unless it is an integrated bracelet, which I have a feeling the OP's likely is.

I don't really see much upside to your plan, but you can give it a try if you're very patient and willing to wait for the perfect watch. One problem is that watches with ruined movements aren't always worth restoring, since they often have other damage and/or missing parts. You're potentially looking at a major project without much potential benefit. In your place, I'd sell your watch and put the money towards a better one.

With respect to the specific question, I'm sure your movement needs a service anyway, so I'd guess that your watchmaker will be willing to swap it at no extra cost while doing the service.