Now the Monarch is currently the head of the Church of England. How can that be compatible with being a Catholic? So if this is introduced presumably the C of E will have to design a new way of finding a head? Perhaps the old line of succession as would have been will continue in the Church, leading to rival branches of the Royal Family? I feel some medieval style dynastic wars coming on.
But seriously, if you are going to be modern and fair about it you would get rid of the monarchy all together. While the monarchy exists, and is hereditary, why not keep the other anachronisms of the position, it is a system that serves us well.