Couldn't this also be a weak flush that doesn't want you to check behind with As or Ks? I think you need to push a lot harder on the earlier streets. Has the min-raise been your standard heads up raise? Mine is usually 2.5x.

I would be a lot more concerned about him leading a flush draw or middle pair on the flop than a T, and your raise gave him the odds to call with either. I'd raise a lot more and call a push. Given the raise you made, I think you have to drop it on the turn. You are either drawing dead or ahead but needing to dodge 12 or more outs on the river.