I am not able to confirm this issue on my end. I just played several rounds of Wii Fit Plus on my Balance Board with no disconnects in sight. It's worth mentioning that the Balance Board only stays connected while playing Wii Fit Plus--if I try to connect in the Wii U Menu, the console drops the connection (since the Wii U Menu has no use for Balance Boards). What are you doing on the Wii U when you encounter this issue?
As Gary kind of implied, the Wii Balance Board is just a big weird Wiimote, like other Wii/U controllers, so it really should "just work". As far as the Wii/U is concerned, it essentially behaves like a Wiimote with a Balance Board "in the extension port", like a Nunchuk, Classic Controller, etc. This is also exactly how the Wii U Pro Controller works, it's a Wiimote with a WUPC "in the extension port", so it would be odd for the Balance Board to be affected differently.