Thank you Randy for this questions and for this thread, because the coral calcification process is as you said "dogma" and most of the valuable scientific information on the topic is somehow "diluted" because of many recent articles dedicated to ocean acidification.
But before go deeper in this first to stress one important fact often omitted even here.
The calcification process in corals happens in a isolated space called subcalicoblastic space. Entrance of any kind of ions in it is almost fully controlled. In this space pH is higher than of ambient sea water and in different species can vary from 8.5 up to 10 in some deep water species and it is pretty stable.
So calcification in corals happens in a controlled environment with pH 8.5 or more and elevated levels of DIC. The one can said OK but then this space is surrounded by ambient seawater with certain pH could influence transport, well no, actually next to calcification space there is another space called coelenteron in which pH is again different than seawater, but this time it is more acidic, has some gradient and could be as low as 6.6 - 7 pH close to the membrane of the calcification space (because of H+ transport from calcification liquid). So
the calcification space in corals is surrounded by water with pH around 7 or even less, and we need to take into consideration when talking about calcification process in corals.
Before you ask for references I'm sure I have some

maybe it worth to make an layman (from layman also) article with some drawings but not sure if I can do it in this forum or I need to have different level membership.
It is even harder to pump out protons, because Ca2ATP pumps are doing this actually from closed environment with pH 8.5 or more to a semi-closed space with pH around 7 or lower, but not in the ambient seawater with said 8.0 - 8.1 pH
Given that it is quite possible
most (but not all) of carbon needed for calcification enter calcification fluid as CO2 (instead of active transport of HCO3 or CO3) via diffusion caused by different partial pressure of CO2 on both sides of membrane.