Same issue on our dedicated server, except a fresh mission (no previous saves). Just hangs when pressing save just like OP. It seems like the command to save isn't even executed based on watching the aliveserver rpt. I could provide the logs, but they look the same as OP with the JSON just pinging every minute(?) or so.
My workaround for now is to roll back to previous version of ALiVE and Aliveserver until a fix arrives. No problems on that version, same exact fresh mission. I didn't get any save on latest version, but rolled back to previous version to test and everything worked.
Just hope ALiVE 1.03 works decently on the new 1.58 update.