The Corps of Engineers has a park on the peninsula:
It is a beautiful facility with nice areas to walk the dogs. Across from it is a cabin rental area. They are very clean. Unfortunatley, I don't remember the name. However, if you call the Corps park, they might be able to give you the name. Many of the guest crafters from Silver Dollar City stay here. Branson Tourism Bureau has all the facilites listed, contact them, too. There are also companies that list privately owned condos - for a group, this might be a consideration.
From the "DogFriiendly.com's Campground and RV Park Guide" three listings for Branson:
Branson KOA 1025 Headwaters Rd, 417-334-7450
Branson Stagecoach RV Park 5751 H165, 417-335-8185 This is a bit out of town
Cooper Creek Campground 471 Cooper Creek Rd, 417-334-5250
Depending how far out you want to go, Kimberling City and Springfield also have facilities.
SDC's new campground does not allow pooches. We looked at the cabins last year (we are Harvest Festival guest crafters at SDC). They are beautiful, quiet and a bit pricey.
Hope this helps,
(Boot Guy, Spur & Strap are still in their bunks!)