I've not been there either. It's been on my "list" for some time though. June sounds like a good time to go. I've been told by several people to avoid the dates early or late in the season as the temperatures can be surprisingly cool.
Actually...June can already somewhat of the warmer side. Keep in mind that the heat index can quite high down there...so...bring enough water and look at the index rather than the temperature alone. Early/late in the year is actually a hit-or-miss. Have been there in March and November over the past 2 years and while it was a bit chilly in the morning, it warmed up to like mid-50s throughout the day. Then again, you may also experience mid-30s and some snow flakes but that is rare in my view.
March/April/May and September/October/November are usually the times I go since riding in like 110+ degree heat index during at least July/August takes quite some commitment...
At the end of the day...Barber is my absolute favorite track by a wide margin in the North/South East...and even on a bad day you can spend hours in the museum...I guarantee you'll love it...
