Lots of good advice here, 1st thing I'd advise... speak to your ISP.
You'll probably need to do some of the other advice here, but then you'll know the options and if there's a cost... no point renting a VPS and having to set that up if a static IP is cheaper, and maybe the ISP can tell you that you're on a /29 and will only have a few IPs to give to the API team...