How can I split this single HVAC zone into two?

(Or otherwise solve this problem)

I have a Lennox EL297UHV gas furnace that is split into 3 zones using Honeywell TH6320R wireless thermostats, connected to Honeywell THM5320R EIMs, all wired into a Lennox Harmony III control panel running about a dozen dampers.

One particular zone supplies two rooms and their Jack and Jill bathroom, a total of 4 registers (2 in the south room, 1 in the bathroom, 1 in the north room). The thermostat is in the south room, which has a large window. Because of this, the north room is always cold. In the winter, the sun warms the south room, the thermostat rarely calls for heating, and the north room temperature drops. In the summer, the thermostat is constantly calling for cooling, and the north room temperature falls.

Since the control panel supports 4 zones, my initial thought was to just purchase a Honeywell TH6320R1001 kit (EIM and thermostat), and put the one damper for the north room on its own thermostat. Seems easy enough, but I'm worried about possible damage caused by having the furnace run for just a single register. Then I thought, if necessary, I could wire the bathroom's damper into both zones, so when either room calls for air, the bathroom gets the air too. That way at least there's two registers in use. I do not have any bypass dampers on this system. With the current setup, the minimum number of registers for any zone is 3. I pulled the cover off the control panel and I can see that each zone has had the PIAB jumpers adjusted by whoever originally installed this system.

The simplest solution would be to just leave the bathroom doors open most of the time. Unfortunately that is not an option. If there's a way to run these thermostats with a remote temperature sensor, or get some other compatible thermostat with that capability, that might improve the situation, but could still overheat the south room.