I don't mean to be brutal, but this hand is honestly kind of a botch from start to finish, I'm afraid.

Firstly, there's no reason to limp preflop with anything here, let alone a weak hand like K4s.

On the flop, you're right, you don't want to lose your customer, but you really, really don't want to lose value, which is why you should be raising a K high flush on that flop every single time, especially when he bets small. The longer you wait to raise, the stronger it's going to look, so your best chance at value is to raise on the flop.

The turn check is really quite bad too, the board is getting even more drawy so you are more and more likely to lose value the further you get into the hand. You need to play it fast on a drawy board like this, to get value out of the wider range of hands that calls you, and to prevent the weaker parts of his calling range from being scared away by future cards. Any 5, 6, 9 or 10 on the river makes four to a straight, and any diamond makes four to a flush. He might even fold a set to a bet on those rivers, in which case you just lost a heap of value.

On the river, I think I probably shove, since our hand is incredibly underrepped and I don't expect him to fold a 7. I'm probably raising any river except a diamond, simply because we're so underrepped, but if the river were different I wouldn't expect us to get called very often. Then again if the river were different I'd expect a different sizing from him, so it would all change.

In general though, probably don't open-limp K4s preflop!