heads up vs. "this guy"

[lane mcbride]

so, I played some heads up stud (high) today, and I had quite a peculiar opponent.
He would call every time, never raise always call. even if he knew he was behind, even if he could not win the hand at all, even if he had aces full (which happened once). He literally called every bet. he never once folded, never once raised.
for example one hand I had (3, 4) 4 a 6 a showing and he called my sixth street bet with king high no draw. and then proceeded to call my aces up down on the river with king high.

so, my question is this. given this type of opponent what range of hands do you raise with on third street, which do you fold, and which do you call given his different possible door cards:
i.e