Raising the privilege to 15 is the easy solution here. 

And since it's Friday evening, easy is all I have the patience for. 

Henceforth, close voting on private betas will be documented as a 15 reputation privilege.

FWIW: almost nobody ever used this privilege even when it *did* work. I counted 25 close votes raised during private betas by users with under 15 rep... In the entire history of the network...