You're probably not the first (nor will you be the last) fursuiter to have thought about and/or done this. There's not much I can say as I've never done something like that before. If you do, then make sure you have a handler with you that can help out with various chores like answering questions about your character, taking photos, helping with fursuit related emergencies, and so forth.
Make sure to get permission from the animal shelter and ask them what they do and don't want you to do.
But I think the best advice I can give is to search out other fursuiters in the fandom who may have done similar gigs and get their thoughts, opinions, and expertise. Try getting in touch with others on Facebook, Twitter, and other social platforms.