I haven't used this myself, but so long as it's a standard Git remote, you can simply use the URL as a remote in SourceTree (Settings in the top-right of the repository toolbar). SourceTree can clone / pull / push with any Git remote that has a reachable URL.
This question has recently come up again via How do i push to Amazon Elastic Beanstalk from SourceTree?, please see the answers from Seth Foss and Andrés Boedo for details, in summary:
[...] you should be able to get what you want pretty easily with custom actions.
In Windows, these can be set up in Tools > Options. For Mac, the tab is in Preferences.
In case anyone wants to do the same thing, what I created was a Custom Action, which runs the script: /usr/bin/git with parameters aws.push
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