What are my options for temperature equalization between two rooms?
Here's a dead-simple idea: use a PTAC unit to air condition the room with excess heat, and dump it into the adjacent room.

That's the general idea behind how companies with huge server rooms recycle heat during wintertime. Their systems are much more sophisticated, of course, inevitably involving heat exchangers, water cooling, thermostats, multiple zones for the heat so conditioned areas don't get too hot, etc.

There are many other ways to do this, depending on the details of your house. If you have ducted heat, you could rig up an air or water-based heat exchanger attached to a new coil in your HVAC air handler. If you have hydronic heat, you could use a similar heat exchanger to put the heat into that system. Etc. It's probably beyond the scope of this website to design such a system for you, but hopefully I've given you some ideas. There are many ways to move heat from one place to another. Get creative!