If you are traveling with someone I've found it to be easier for one to go get the rental while other waits with pets/luggage. Then pick up from airport and only load one time and not need to load, unload into bus and reload into rental. If traveling alone, you could check with curbside porters to see if they could hold your pets while you get your rental. I would call the Budget office directly and check with them to be sure prior to arrival regardless do you know your options.
Thank you for your service and welcome home. I have seen service dogs in the shuttle, and cats in their crate. I do not believe your pets would be a problem. You could double check with airport information and customer service.
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