I am wondering what kind of expert can get TO what's making the noise, whether it's a critter or something that got stuck that thumps. From your description, I'm not sure that calling Animal Control is the right thing. They don't have the expertise to get into your walls or your "stack". It needs to be a specialist, to prevent your house from being damaged by someone who doesn't know what they are doing. I'm sorry, but my house would come first over the animal, if indeed it even is an animal.
I just don't know what the "stack" is and what it does, other than it has a pipe in it. It doesn't sound like you're talking about a chimney, and I can't figure what the pipe is for. Anyway, I sure hope someone else has some advice for you other than Animal Control...
Dawn