You have another option.
The Owners Corp is required to maintain common property, it’s true, but you have a choice. Yes, you could fix the Air-con or you could tell the owner that, as there is no record of permission, common property is going to be returned to its previous condition (no air-con) at the OC’s expense.
If the owner wants to keep the air-con, they can agree to a special resolution by-law giving them responsibility for the air-con and its maintenance and that of the common property around it.
Or they can arrange a time for your builders to come in and remove it and reinstate the CP.
Just keep the wheels spinning in what promises to be a long, hot summer and I suspect your neighbour will soon come round to a reasonable compromise.