I may get flamed for this but here goes.....
I'm not a coon hunter but my cousin and his dad are hard-core, competition coon hunters. I will add that their dogs will rarely, if ever, run a deer but some guys dogs probably will.
My observation over the years on our family property (1,200 acres in McNairy county) is that their hunting has had zero impact on the deer hunting, even during the deer season.
My opinion is that if someone is a turkey hunter, you might try to develop a relationship with a good local coon hunter and give them permission to only hunt specific times during the year with the provision that they shoot out every coon they tree. I imagine they'll honor the dates you've allowed them to hunt rather than cross you and loose the permission....and that it will help your turkey population.