I would be a bit concerned that we might go a lot off in 4C if partner has doubled on a 4162 shape and is planning on correcting 4C. On the other hand I suppose 3H might make but that seems very unlikely to me.
I don't agree that after the double 1H is the normal action. Zia would shoot any partner that bid Txxx after a double. Personally I don't see what is wrong with XX (as a passed hand) or passing and doubling later.