Few summers ago, whilst working for RYR, I had a similar situation. On stand ready for boarding but no ground conditioned air available and a limitation on APU usage imposed by airport regulations (no longer than -5min prior to off block). OAT around 35 degrees Celsius. Refused to start boarding process until either ground conditioned air available or authorisation to use APU outside the airport regulations received. Kept pax in a nicely airconditioned terminal until things were sorted out, causing a significant delay.
At the end of the duty wrote a small report explaining circumstances and decisions made. Was backed up by the management. No stress and no management pressure. It is a captain's call, even in RYR.