If you were going to build a proper STOVL carrier you wouldn't start with an LHD hull. If you want a ship which can function as a reserve for your single STOVL carrier when the carrier is hors de combat, e.g. in refit, then IMO it's an excellent design, a good compromise. For that use, the Spanish navy intends to seal the dock, & install additional equipment to support the carrier role in the vehicle deck, for the duration of the carrier tour. The extra kit would be stored on shore normally.
I think a proper STOVL carrier should be faster. More wind over deck, for a start. That would mean a change to the hull (the stern is shaped that way for the dock) & propulsion, probably to something more like Cavour - so you might be better off starting with Cavour, & seeing how she could be improved, or changed to meet your requirement, rather than starting with the BPE design.