You should not need a permit for this project. Unless the city has some weird rule.
I would avoid the sump crock and utilize the method you have drawn. (No worries about failure and much less cost) It will def suck getting around your HVAC equipment, but since you have the appropriate drop in drainage line, seems like the better path to take. The studor vent would be a much easier option and less plumbing to run, so I would vote for that.