Staging sync should also trigger publish on staging site

[ahouseholder]

Describe the bug

We have an action on the staging repo that syncs with this repo on a schedule. It should trigger the deploy_site action in that repo as well, but it doesn't because it doesn't have the right token.

Expected behavior

When the sync workflow pushes to staging/main, it should trigger the deploy_site action.

** Workaround **

Current workaround is to manually trigger deploy_site after a sync when desired.