A fairly easy approach leads you to the base of an obvious sandstone cliff. There are several bolted routes ranging from 5.8 to 5.10. Some shady spots and reasonable temps most of the summer. The approach is a little dicey (steep and loose). Good approach shoes are recommended.
The campground is in the Gallatin Canyon approximately 48 miles south of Bozeman, MT on US 191. It's about 7 miles south of the Big Sky ski area. Enter the camp ground by crossing the bridge. At the fork in the road veer right. Continue back until the obvious trail head. The cliff is just up the trail.
The Red Cliff's Campground is open for camping from 5/15 - 9/15. Before or after this time you have to park on the road and walk into the campground. This will add a bit to your approach. You can also park in the pick nick area and walk in.
It very well could be. The only limestone I have climbed has tended to be a darker gray color. I suppose I have seen some that is more of a tan as well. I read somewehre else that this was sandstone, and looking at the rock it was very red. However, the ground there was saturated with what looked like red clay. So as you point out I suppose this could be limestone stained by the red clay.
The rock formation did have a lot of pockets typical of limestone, so it is very possible.
FWIW, there is a such thing as red limestone. Iron oxide in surrounding soils can turn the limestone red, yellow, or brown, whereas carbon impurities will turn it the typical blue, gray, and black. Often combination of these minerals streak or stripe the exposed cliff creating the "black & tan" and blue & gold combination that are so appealing. Think Black & Tan cliff outside St George, Utah or the French Cattle Ranch at Tensleep, Wyoming or even the walls of areas like Ceuse, France or limestone crags in Spain. Red & brown limestone is not unusual to find in areas such as Montana and Wyoming.
FYI there are 2 boulders with at least 5 established and cleaned problems VB-V2+? not amazing. To access head south from RCCG 1/3 mile to a dirt road into forest drive back 1000 feet to dead end near forest service cabin from here you cannot miss them. If you hike the trail back 30 minutes 2-3 other non cleaned boulders on the right appear along with what may be Big Sky's best limestone walls if they were cleaned and bolted.