I will happily design, for a small fee, a system for you that sends you the position of a hypothetical satellite based on a simple time calculation. It's a trivial task to anyone with a basic understanding of programming.
The fact that a piece of hardware does what it's been programmed to do doesn't prove much about the Earth's shape.
Only problem with that idea is that it won't work.
Let's for the sake of argument that a ground based GPS tower is pretending to be a satellite in orbit, first the position being transmitted will be continuously changing, and the ranging information from the time sequence will need to be changing as well. I can't see that this would be a difficult task, the problem is that the receivers in the transmitter coverage area will not be able to get a correct location fix, since the transmitter itself isn't actually moving, and the ranging sequences can't be simulated for multiple gps receivers at the same time.
The way the GPS system works, requires the transmitters to be exactly where they say they are. If they aren't then the receivers can't fix their location correctly. It's worth noting that they don't actually have to be in orbit, but they do have to transmit their real location. You could have your own local GPS system with a transmitter up in a tree in your backyard, and another in your back pocket. It will work just fine, so long as the positional information being sent is true.
The actual transmitter locations can be determined by looking at the raw data stream, when you do this you will find that they are in fact in orbit where they are supposed to be.
If someone wants to get into building their own GPS receiver and verifying what I'm saying, I'd be happy to help if I can, It's not that hard, or you could purchase a suitable receiver chipset that provides access to the raw data. This one is NEO-6T based for $49.99 and does cm precision and can use the open source RTKlib, so you can look at GNSS raw data, and verify the position calculations that show the earth is a globe if that's your thing. If it's not then cm level gps accuracy is still nice to have.
http://www.emlid.com/neo-6t-precise-timing-and-raw-data-receiver-for-rtklib-for-49-99/The RTKlib software can be downloaded here.
http://www.rtklib.com/Happy GPS transmitter hunting.