A little more progress, it seems the Sublime SFTP is waiting for the input prompt to be "Password: ".
By default, google authenticator’s libpam prompts for "Verification Code: " so sublime never picks up the correct prompt.
A very hacky workaround is to configure google authenticator to prompt with "Password: " instead.
auth required pam_google_authenticator.so [authtok_prompt=Password: ]
And then sublime SFTP properly picks up the prompt and asks you for your MFA token.
This obviously only works if you don’t use regular passwords as well. It would be nice, though maybe also a bit hacky, to have sublime SFTP configurable to detect the prompt, so we could configure this on the client side instead.
I think ideally, sublime SFTP should detect when input is requested more generically, and prompt the user appropriately though.