Theres no way he would fold KK on the turn when he has the nuts full house

) Even on the flop it would be a hard fold

)
And it was FL anyways. Calling a 3$ reraise anyday

)
A good player would certainly not fold it. Some unknown who has been chasing the BBJ for 3-4 nights straight could easily misclick or timeout, or get disconnected or have a heart attack or his bot might stop reading the cards in the middle of the hand (it's hapenned before), or his profile might fold due to a stack misread. I'd simply go for one raise/call when playing a potential BBJ hand in limit. In NL I'd just shove vs the first flop reraise

I think FL is the best way to chase the BBJ, IMHO. the play is stacksize-independent so you don't have to worry about bigger stacks pushing you out. you can afford to chase straight flush draws and quads all day along for a few blinds. You just have to make sure that at least the turn and river are bet.