Since this is a Mendix Git repository, you can avoid downloading branches as plain copies by cloning the repository with Git metadata included.
Instead of copying the branch files, use a proper Git clone:
git clone --branch BRANCH_NAME --single-branch https://pat:YOUR_PAT@git.api.mendix.com/APP_ID.git
This keeps the Git information, so the branch remains a real Git repository and you can continue to commit, pull, and push changes.
For multiple branches, you can automate this command and create separate clones per branch. Each clone will still contain the .git metadata and behave like a normal Mendix Git project.
Kindly mark this as the accepted answer if it helps.