To use different shell material they would no longer be coral. Breeding corals with different skeleton material is like trying to breed humans with a different skeleton material.
For context: Many shelled organisms (cnidarians - corals, brittlestars, sea urchins; molluscs - shellfish) build shells from CaCO3, either as the polymorphs calcite or aragonite. Other options for shell material are silica (SiO2; e.g. some radiolaria, siliceous sponges) or chitin (C8H13O5N; e.g. shells of crustaceans). Animals make their skeletons from hydroxy apatite Ca5(PO4)3(OH).
There was recent research that coral adapted to cope with changing pH[1], but weren't adapting fast enough and the symbiotic algae that bound reefs together did not seem to adjust at all. Plankton is apparently equally vulnerable to acidification.
The unknown for coral is the rate of change of temperature.