You may need to rethink your approach a little.
In general, the workflow of
Make changes in database -> Update Database Project -> Commit changes
to Source Control
is not well supported by SSDT; in particular the part about updating a project based on changes to a database.
If this were a .NET project, would you be patching the binaries on the server using a hex editor and then decompiling the results into a csproj and associated cs files to store in source control? This sounds ridiculous, but it is analogous to the workflow you are suggesting for your database projects.
I believe the Redgate tools - with which I am not particularly familiar - have some support for updating source control from a deployed database. I am however familiar enough with said tools to know that the intended use case is not
Make changes in production -> Update Source Control
IMV, You should probably be looking to solve the "source control" and "audit" problems separately.
To do this (with SSDT), you only need to update the database project manually once, and add the resulting files to source control.
After that, you can make changes in the project first, commit them to source control, and then deploy these changes to your database. This process is easily automated.
Presumably it is only a subset of the data in the database - the "static" or "reference" data - that you need to store in source control? The most common way to do this is using post-deployment scripts in the database project.
Regarding audit, you have a couple of options. Given that the history of your "deliberate" changes will be in source control, the main concern of audit is detecting uncontrolled change in production. This can be done with database triggers, or, I believe, by some commercial products (that generally use database triggers behind the scenes). On detecting such changes, you then have a couple of options - roll back the change, fire the DBA, update the files in source control, etc, etc. I'm not sure it's sensible to automate this part of the process, as you will probably want to consider why these changes have occurred.