I suggest you look at this ruling by CTTT as it is very similar to yours.
Basically the ruling was
1. Permission is hereby granted for the Applicant to keep a “bichon frise” dog on the subject lot provided the dog is kept wholly within the lot at all times, does not interfere with other occupiers reasonable peace and enjoyment of their lots and the common property. The dog is to be carried over common property when being transported to and from the lot from time to time.
2. This consent only applies to the keeping of this particular dog and not to the keeping of subsequent pets.
These conditions are pretty standard and so long as you are prepared to abide by them you should win your case.
If the bylaw had said no pets period you would have had a more difficult fight.