Okay so I tried this yesterday. Turned off Background App Refresh and signed out of both Teams apps. I signed into my Macos device first.
Everything seemed to work yesterday. Overnight the issue appeared again and now this morning I am unable to change my status on my mac and it is set to "Away". The only way to change this on my MacBook is to actually open up my iPhone and activate teams, which then changes my presence everywhere. No matter what, my iPhone dictates my teams status everywhere.
In summary: turning off background app refresh did not help (great idea to try, however!). It also doesn't affect calling or notifications from being delivered. I see no discernable difference when background app refresh is on vs off on the ios app.