I see the list on the API documentation matches the list you have for this action. I was unaware that php5-fpm would restart PHP FPM in general, I guess I could try that.
I haven't experimented with other values against the API directly. I only need to restart this one service on deployment and found this action which looked like it would work. I noticed the php8.0-fpm service name in the AJAX call when restarting the service manually using the UI.