I have a problem here. Im wanting to put down stone tiles on a plywood deck. The problem is I have a steel security screen door and not enough clearance for the door to open and close. In fact in one place I have "maybe" 1/4" at tops! There used to be carpet on this deck but I ripped that out.

I want to go with stone tiles, but think I may have to compromise and do vinyl just because of the thickness/thinness.

I have no aversion to doing a little concrete work, maybe with one of those expensive high strength ones like Cement All just so the door would clear the deck. My concern here is if reinforcing or backerboard would be required as it would be too thick. Can I just concrete thinly on the wood and build up to the height of the tiles I want to put down?

Ideally I would like to be able to cut down my security door to fit and add a doggie door at the same time......... Help?