Tough spot here, I hate to lose big with KK if money isn't going in preflop. He was SB and first to call your preflop raise, and now he's not respecting your pot size bet and by check/raising he's representing a strong hand.
About the range you're putting him on, although possible, I doubt he has Ax hearts. On $10 max tables players tend to check/call with that type of draw when they're first to act. The problem with these levels is you can't really exclude he has a hand like A3s. An overpair (not AA) is also very likely. The 55 rare but possible. I just don't think he's bluffing
, he has something and you are either way ahead or way behind.
As to your question whether to push or reraise, I'm not sure but I think it means you are ready to put your stack in this pot because you're almost sure you're ahead. I think he will call both a reraise or a shove with the range we put him on except maybe if he has a flush draw which I doubt, so it doesn't matter which way you go.
If, instead, you are asking about reraising for the purpose of finding out more about his hand, you need to think ahead. You can't reraise because it will make you pot committed so whatever info you get is pointless.