The Provari is 4.1" L X .90" Dia and uses 18500 batts without the extension. Couldn't find any specks on the BB Size but it uses a 14500 without the extension so same length
batteries. The Provari does have electronics built in which will add a little length but not that much. It's a actually pretty small for a VV Tube a bit on the heavy side though if that matters to you.
Size aside these don't compare, a regulated vv mode with digital adjustment and readout compared to a 14500 tube, although a nice one. Get the Provari and forget about it, the size difference won't be that much and you'll get one eventually anyway.
There are a lot of small tubes out there cheaper than the bb, if you really have to have one get one of those which will save you some $$ off the BB and get both but get the Provari, you'll be happier with it in the long run.