It looks the action is hardcoding the expected path to download the local tunnel to. This causes a permission denied error if you are not using a filesystem setup exactly like a Github hosted runner.

https://github.com/browserstack/github-actions/blob/master/setup-local/src/binaryControl.js#L47

The code should probably use one of the Runner provided environment variables for where you can download tools to. Either RUNNER_TEMP or RUNNER_TOOL_CACHE.

https://docs.github.com/en/actions/learn-github-actions/environment-variables#default-environment-variables