Claim Draw by Repetition
It seems pretty clear your opponent was just using taking advantage of the clock, i know the clock is part of the game and you have to manage your own time but in a clearly drawn game it is very unsporstmanlike to just outrace your opponent on the clock, just my opinion but you are right you should be able to claim a draw when the position is a dead draw.
Had I taken a pawn, he would have queened.
Ironically, your best bet was to take a pawn in the hopes that your opponent was trying so hard to stay ahead on the clock he would have moved his king again before noticing and then you win easily.