We would stay at Miller Manor (340-774-1535) when traveling with Smokey (RIP) and have to overnight in STT on our way to Tortola. They were the only place we could find that was in my price range (also inexpensive) and I do know that they rent long term. The last time we stayed there with the cat was in June of 2013 so I don't if things have changed. Marge and Harry are great hosts and always made us feel at home. Their website is
www.millermanor.com. Most places will not take pets. Hope this helps.