I would use a self-hosted runner for that [1]. You can setup the SDK in that machine and it will be available for any jobs that end up running there.
Depending on your requirements/scale the runner(s) can be a VM in your main machine, a cheapo dedicated server (hetzner/ovh) or even autoscaled hotspot instances in your preferred cloud.
It is less pure than using GH's runners and having an end-to-end setup/teardown for your whole toolchain, but it would work just fine. Definitely better than without any CI.
It doesn't seem too hard.