Hi!
Never coldcall a single player limping,pretty much a golden rule for FL... raise or fold spot!
In this case raising is clearly the best play,trying to isolate yourself with a bad player. And even if SB,BB call you have a very strong multiway hand and dont mind playing a bigger pot.
On the flop raise is very well played.You want to charge possible gutshots,you want possible further action vs Kx or maybe induce lighter calldown as your hand clearly looks weaker when you 3bet flop than when you raise turn.
BBs 3bet looks somwhat scarry but its quite obvious that AA,AK,KQs or even maybe KJ,QJ,JTare more likely than KK,AT ,maybe KTs. So you definatelly have to go for another raise somewhere.You can just cap the flop but waiting for the turn is fine as well.
This turn is not rly that bad since not of the before mentioned hand didnt rly improve except JT...I would most certainly raise like you did. His 3bet is scarry however,now AA,AK get way way less likely and you are basically hoping for an overplay or spazz....calldown
wp,imo ,well except preflop which is a clearcut raise!!!