Hi @pbrewton, welcome to the forums!
There is no way to send a checkout request to a POS application via another device (like the terminal API), unfortunately. We have a POS API and Reader SDK that allows you to build your own mobile POS that can take in-person payments like Square Register, though.
With that said, the Square Register actually doesn’t allow other apps to be installed on it, so POS API nor Reader SDK will work specifically for Square Register. There’s no way to do this at this time with Square Register, unfortunately.