So you have a database that is saying it is corrupt, you have no certain way of knowing that you are not making the corruption worse because you are still updating it, and are worried about the downtime while you salvage what data is left.

My advice is to bite the bullet and schedule some downtime where you can copy out of your database everything that is still available.

You should also work out scripts that can check the new database for application integrity. The declared RI you have can deal with physical integrity and warn you of child records with missing parents, etc. However declarative RI cannot warn you if a parent has no children when your application expects that it should have.