Sawyer was heroic all through the last couple of episodes in Season 4. He saved Claire, and was most solicitious of her welfare. He also warned Ben not to hurt one curly hair on Hurley's head. He also went back with Jack to search for the stragglers who would otherwise have missed the boat (literally).
Then he jumped out the copter to save some weight. I think being around Aaron and Walt a lot on the island has probably made him wonder and care about his little daughter; but he realises he hasn't got off to the best start with her. I think THAT is what is worrying him (nothing to do with Kate). It's almost like he is sending Kate to pave the way for him.
I wouldn't be surprised if Kate and Sawyer end up together. They both seem to have got very maternal and paternal lately whereas Jack, although he was reading nicely to Aaron, seems to doubt that he is good father material.
I think initially Kate thought Jack was the more 'settling down' type but after Sawyer's display of paternal caring for his daughter, she may start to have second thoughts.
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