Your coop is too small for birds that are that big. A minimum of 10 square feet per adult bird is needed. That is clear area that is not used up by anything else such as feeders or waterers.
They need an even bigger area for a run.
Here in Wyoming the wintet temps can and do het to -30⁰F. My turkeys spend all their nights on their roosts outside. Their roosts are sheltered from the prevailing wind.
It's a pecking order / territorial dispute. It will take time for them to work things out.
For the health of your hen turkey, it would be best to add more hen turkeys. A tom can wear out a single hen.
Turkeys and geese or any waterfowl are not a good fit. idea. Waterfowl love water and...