If your opponent has a forced perpetual check you'd have to take the draw.
endless checking

This is part of chess, if you have a better position, or more material, it is your responsibility to make sure that he can't force a perpetual, or a mate for that matters.
After the same position has opccured 3 times, either of you can claim a draw.

If your opponent has perpetual check, you are not winning. Your opponent has a forced draw by repetition. He should offer once; if you ignorantly refuse, he should claim the draw on the third repetition. In your opponent's shoes, I would use premove in such a situation to get to the third repetition quickly.

what would you do if you're ahead and winning in a game but your king is exposed and has nowhere to run and remain unchecked. your opponent can check you endless amount of time hoping for a draw (and does) but he can't mate and on top of all that he offers a draw every 2.nd move?
would you accept the draw? what would you do?
His offering a draw every other move is rude. But other than that, he's doing nothing wrong. Do you expect him to just say "Oh, I guess I'll resign" when he can get a draw?
If you can stop the perpetual checks, then stop them. If you can't stop them, then you're going to end up in a draw.
You say you are winning, but if you can't stop the perpetual checks, then you aren't.
what would you do if you're ahead and winning in a game but your king is exposed and has nowhere to run and remain unchecked. your opponent can check you endless amount of time hoping for a draw (and does) but he can't mate and on top of all that he offers a draw every 2.nd move?
would you accept the draw? what would you do?