You can publish straight to IIS, you know. You don't have to publish to folder and then copy to IIS. And why would you not want to test against IIS if that will be the platform that eventually hosts your Web API?
When you install Visual Studio, you also automatically get IISExpress. This let's you do some basic testing, but be aware you are going to be bypassing a lot of security and permissions issue that you will sudden discover when you deploy to IIS.
So if you are really trying to avoid that extra deployment step while testing, go ahead and just run your code on your dev box and have the phones point to your dev box, but be aware that you'll still need to do more testing later.