>Testing an HTTP API involves running the server, and running a client against that.
That's exactly what tools like Postman are used for. How are you doing any testing during the development process if you're not hosting your web API even locally?
>I haven't tried it, but I think it'd be quite awkward to get a server setup from Postman
There's no need to as I mentioned that you're going to be hosting it somewhere during development anyway.