Author here, if anyone has any questions in relation to me or Sublime HQ please feel free to ask.
I contribute to git.git, and it would be interesting to know if there's inherent issues stopping you from doing that, or if it's implementation problems in some cases (e.g. missing plumbing commands or features). There's definitely interest from upstream in reviewing patches / helping if there's missing or inadequate plumbing.
You'd get upstream features for free as they come along. E.g. presumably you haven't implemented the new MIDX format, but that speeds up pack file access by a lot for some use-cases, and presumably the boring bits of low-level git operations aren't much of a selling point in and of themselves.
Aside from whether you'd use "git" itself, such a trick of using a slave process you'd talk to over IPC of some sort would cover some of the issues you wrote about, e.g. issue with sharing global state with libraries like Breakpad.