There are two NF campgrounds around Frisco - Heaton Bay (39.605, -106.07778), and Peak One (39.58472, -106.06944). Peak One has many sites available for two nights, beginning July 28. Also, many available the week before.
Non-electric, but it doesn't get sweltering in Frisco. If you have a powered vent you can leave open, along with a source for fresh air, your cats should be fine for the day.
There are a couple more campgrounds on Swan Mountain, just east of Frisco (Lowery, 39.59861, -106.03444, and Prospector (39.59583, -106.03444), but the road up from Frisco is steep and winding. You would be ahead to approach from the east, using the Dillon exit.