I have confirmed that POST /cc/api/source/update/id has been deprecated. It has been replaced by the equivalent V3 endpoint as reported here in Non-Public API Deprecations.
As you and others have discovered, it appears that the ability to update useForPasswordManagement may not exist with PUT or PATCH v3/sources/id.
I see that you have opened a support ticket. If the support team is unable to resolve your issue and determines that this is a bug, an appropriate engineering ticket will be opened.