No, the Beach Club bus stop is not currently under construction. I have not heard anything indicating that there is upcoming work on the BC or YC bus stop. There is a refurbishment of the Convention Center happening now, and that impacts the YC parking lot area and the resort bus route, I believe, when the bus travels between the two resorts.
When I was out at the bus stop to check for early morning buses in August 2015, I saw buses for both MK (9 am open) and AK (8 am open) shortly after 6:30 am- I wrote about it in a post I have linked in the first post on Page 1 (search "Bus Stop Babe").