I have a workable solution thanks to everyone. I am using both RemoteRig boxes. It took a bit of effort to get them to talk to each other between the house and shop, but I got that sorted out. I basically started with the K3 Remote manual from Elecraft and made a few changes.
There does seem to be an issue with the P3. When the P3 is connected to the K3S and the RemoteRig box is connected to the RS232 (PC port) on the P3, the P3 comes up normally, but the K3/0 Mini doesn't "connect". The power button works, but nothing else. It continues to say TERM in the VFO B display area. If I hold the Menu key for 5 seconds, it goes to NORM mode and then if I hold it for 5 seconds again, it switches back to TERM mode and everything works.
If I leave the P3 off, the K3/0 Mini starts fine. I also tried connecting the RemoteRig box directly to the K3S and the P3 to COMM 1 on the RemoteRig box. I tried several different settings for COMM 1, but the P3 would never get the frequency information.
I don't plan on streaming the P3 video at this point so I'm just going to change the Power On configuration of the P3 so it doesn't come on when the K3S is turned on. I don't think the Y-Box from N6TV would help in this situation.
It took a bit of work to get a straight key to function, but I found a post on the RemoteRig forum that helped with that.
I think the only issue I haven't solved is some noise coming from the K3/0 mini even with the volume turned down. This noise goes away if the network cable is disconnected. The volume also seems to be less if I use a shorter network cable. I'm hoping the Wifi option solves that.