I had to do some additional work from Patoshi’s answer.
In Tools > Options > General > SSH Client Configuration, I had to point it to the id_rsa key, and select OpenSSH as the SSH Client (Putty/PLink is selected by default).
Then each time you start SourceTree, a prompt will appear asking for the passphrase to decrypt the id_rsa key (assuming you created it with a passphrase to encrypt it). This looks like so:
At that point, I could go to the Clone page, paste in the SSH address from my Gitlab project (git@localgit.local:Blah/Whatever.git) and it cloned it. No extra prompts for username or password (since it is using SSH keys).