Yes, temperature has increased with CO2 in the atmosphere. However, the temperature increase also follows the lagged response from the sun. Both variables likely have a significant effect, and I would claim that the solar lag changes are much greater than the CO2 changes.
Has it? We really don't know for certain. I doubt it has as much of an effect as you believe, since the total dissolved carbons in the ocean maintain an equilibrium based on temperature and salinity. I will contend that the drop in pH we see is primarily due to SST increases. I'm not discounting CO2, but don't discount the buffering.
I wish I know the paper title. Long ago, I came across a paper probably about 40 years old. It had graphs on the ocean pH, by year. The pH of the ocean was it's lowest. Even lower than today, about 1958. The same year that we had peak solar radiance.
Remember, when it comes to SST, the solar radiance of the sun is almost fully absorbed at spectra shorter than 1 micron, at depths deeper than longwave. The longwave from CO2 is mostly absorbed in the first micron of the surface, and the evaporation process in enhanced by this. This depth is evaporated rapidly, and even takes heat from the water just below it. I will contend that CO2 causes insignificant ocean warming in any areas that matter for coral, because these already warmer waters evaporate at a rate faster than the CO2 spectra heats what remains.