I'm in the process of moving a site to local (MAMP) dev environment that has not been in BitBucket or git before and am using Tower App for git responsibilities.

When I copy a site from the live server down to my local, Tower (understandably) sees these files as new and needing to be committed.

What I'd like to be able to do is copy the site from live/production server down to my MAMP server and have Tower not see them as new, as I don't want to push the entire site to BitBucket repos and then onto the live server.

Any recommendations?

Git Repositories should include All Your Code

Therefore not committing files your code depends on is strongly discouraged. You'd end up with a partial copy instead of a repository in the literal sense.
This also means your entire site needs to be pushed to bitbucket.

Reducing setup time

If I understand your workflow correctly, you want to minimise the time needed for push and deployments. For the initial push to bitbucket there is no way to speed things up in your setup. However, most deployment tools offer you the possibility to only send changed files to the live server. This is a configuration issue and best answered by your tools documentation.