After adding the GitLab.com account in the Preferences I am continually asked for password (sometimes several times in succession) even though I always enter the correct private token that I created specifically for Sourcetree.

If I had to guess, I would say that it likely has to do with how the keychain is used from the Sourcetree app.  Since I also have my gitlab.com credentials stored in keychain for the browser to use, it is my guess that there is a collision in the Sourcetree app when it tries to save the credentials to keychain.  Sourcetree is going to need to store them in Keychain using some attributes that will not collide with any credentials already stored.

Of course, this may not be the case, but at the moment, using GitLab.com account with Sourcetree is not working because the Sourcetree app is constantly asking me to re-enter the "password" (even though it needs a private token), and does not remember the credential even though I check the box.