As far as I know, (the locals will correct me if I'm wrong) pools are basically non-existent north of Calumet in the winter. Hot tubs are fairly scarce as well.
Over the years I've stayed at the Mariner North, The Pines, and the Keweenaw Mountain Lodge in Copper Harbor but my first choice by far is the Lac La Belle Lodge in Lac La Belle (about 10 miles from Copper Harbor if you didn't already know).
The LLB Lodge (laclabellelodge.com) has cabins in your price range, gas on the premises, a great bar and outstanding restaurant for lunch, dinner, or a late evening hunger attack. They also have a pretty well stocked store and off-sale. They don't serve breakfast but the cabins have full kitchens if you're inclined to cook something. There is no pool or hot tub but that is pretty much the norm up there. I've found the variety of their menu to be more than adequate for our stays but the Mariner and Keweenaw Mountain Lodge also serve food. I think the Pines may serve some food but I can't say for sure.
Hopefully this will be helpful.
(I forgot to add that the LLB Lodge is literally on the trail)