Hmmm...a little surprising you're having so much trouble. Try going down Sai 3 and 4 a bit (going away from MUIC and towards Phetkasem Rd.) <<I think that is South.
There are several neighborhoods between these two Roads (Puthamonthon 3 & 4) Some used to live on Sai 2 but I'm not as familiar.
One popular place is MooBan Preecha. It's on Sai 4, about a half a click past the Queens park on your right.
Much easier to shop for houses if you have a car, but I did it before buying one by using taxis and then legging it in these neighborhoods. You've gotta be active, talk to people if they're out and about....leave a card with your number at the gate of empty houses.....post something near the neighborhood convenience stores...talk to those people, etc....
These houses will have ZERO problems with pets,....and you should get a decent yard!