I live in a project where we are on top of this. Condo Byelaws explicit for no short-term lets, but the biggest factor is enforcement and here we are fontunate to employ our office staff directly rather than using a management company. Providing you can get enough support, then an AGM resolution to change the management company to your own staff could well make the existing company much more proactive in enforcing the rules (I am assuming your rules do not allow short term lets here).
I would look to go the own-staff route in any case. Talking to others who live in projects with Man.Cos, the difference is marked - they work for us, not for the Company we employ.
PH