Thompson's Lake State Campground is the only place close to Altamont. I have been there a lot when I was younger and it is a very nice spot. There is an interesting nature center across the lake that the park supervisor can give you directions to. From the main entrance to the fairgrounds go left through the village of Altamont, bear right and go up a long hill about 3 or 4 miles to route 157 and take a left. Continue for about a mile or so and you will see the entrance to the campground on the left.. I don't know what the fees are as I have not stayed at any State sites in a while. Thompson's Lake is nice but not very isolated. There are a lot of camps and cottages around it. I'm not sure but the organizers of the Scottish Games used to allow camping in the parking lot of the fairgrounds if that is another option worth looking into. With the holiday weekend and the campground's close proximity to the cities make sure that you check and see that it is not already full. Good luck and enjoy the games. ( I will be loading up on Cameron's meat pies).
Thompsons Lake campground for certain. I had thought there was also a small campground off Cole Hill Road in Berne and/or one around Warners Lake - both would provide good access to Altamont.