feature: Add automated benchmarking to GitHub

[kshyatt-aws]

Issue #, if available: N/A

Description of changes: Split off the benchmarking logic from unrelated stuff in previous PR

Testing done: Worked locally with act but needs live test on GH to make sure

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[kshyatt-aws]

Looks like we need a gh-pages branch no matter what, perhaps we can create an orphan branch for that? Or host our Sphinx docs there?

The base branch was changed.