I have taken my Medic Alert/Service Dog into apartments where the people did not have animals. She has been certified as a therapy dog with Lend A Heart, she is registered with our county as a service dog and medic alert dog (saved my life 3 times this month alone...with witnesses on one occasion), and ... I can't tell you how many classes she has been in. My wife is her primary trainer.

Even the hospital let her in my room during a stay and let us into my wife's room when she had her knee replaced.

I never heard of an apartment saying no. I guess they can try. If the general public is allowed in one of their apartments (a party or open house), then the service dog should be allowed.

That said, if your dog is poorly trained, etc. the mgt has the right to ask you to leave and maybe even request payment for any damages your dog might cause. Most real Service Dogs are NOT a problem anywhere they go. They are welcomed back, at least mine is. I have seen a couple of people who said their dogs were Service Dogs, but the behavior of the pet was questionable, with growls at people, etc.

Good luck with it....
PS-- like you, if the tenant seemed concerned, I would not take my dog in either. I try not to offend or irritate people most of the time.