Knowing nothing about the player this is what I would do. I would put the big blind on
and the button on a pair of 10s or a made straight. So I would bet if I though there was a better chance of the button having a 10 and check/call if I thought that the button had the straight. Deping on what I have seen him done in past hands.
Also I would have raised on the turn to try to get the button out of the hand and go heads up with the BB to the river.
What happened in the hand?