The degree to which the word shelter has negative associations for some people is not something I would see as deal breaker territory.
I will add that there are far more uses for shelter than just homeless shelter or animal shelter. Just a few examples:
A book about a New York woman's house hunt was titled "Gimme Shelter." (Author: Mary Elizabeth Williams)
The basic essentials in life are typically listed as food, shelter and clothing.
Shelter from the storm is a well known positive expression for sanctuary during a crisis.
Shelter magazines is the umbrella term for magazines having anything to do with homes, such as home plans, bed and bath, kitchen makeovers etc.
The format "(noun) shelter" is similar to and might more readily bring to mind "animal shelter" and "homeless shelter" than a phrase in the "(verb) shelter" and "shelter _____" formats.
The mind is very pattern-happy after all :)