Preflop
The first thing to always look at in a tournament poker hand is stack sizes. In this situation you have 3 players with rejam stacks behind you, and you have the chip leader on BTN, which is the best seat on the table. For both these reasons I prefer to let a marginal hand like KTs go from this position.
If you are going to play, you should size down to 2,5BB or even a mini-raise, so that you lose less, when you have to fold to a 3-bet. It is also nice to already think about, who you are going to call, if they 3-bet, and in this spot that is probably only SB at most, since he has less than 10BB.
Postflop
You could just call his mini-raise, but turns and river can get a bit awkward, if you dont improve. Getting it in takes away his positional advantage and his opportunity to "bully" you with his stack, so I am totally on board with this. Of course you will sometimes be behind, when he call you, but that is baked into the math, and 40%
equity is certainly not the end of the world. Just sad that you were unable to improve on any of your many outs, but thats poker.