We often would want to go with an overbet to attack a capped range. Given the action I don't find that his range would be capped. BB has a lot of 2p+ combinations he can easily call.
If we would construct an overbet range I'm not sure if we would want this hand. KTcc is way better for example. Blocks JT, KJ and has a gutshot to the nuts or second nut flushdraw.

I would prefer not splitting range and have 1 betsize when we decide to continue barreling on this turn.

I see this a lot: over betting with super high equity hands to try to get maximum fold equity while having 12+ outs, shoving as much money as possible into the pot and not wanting to see more streets. It can't that terrible of a play due to raw equity, but are you overbetting J9/JJ/99/33/AA on this board? Where's the actual cutoff in your range where you stop doing this? Cuz if im playing this style then im just going to overbet with the nuts and combo draws. If your not doing this with hands like top set as well, then I don't understand how you can construct a sensible range.