Considering I'm doing my own "perfectionist completionist" run on Ultra Moon, I didn't even have room for all the different forms of Pokemon on that game. With three spaces left over and excluding most Unown and Furfrou forms and probably some others too, I got to 963 total Pokemon. Doubling that for shinies gives me a total of 1926. Now if you include all Pokemon that I didn't, that would likely bring the total to over 2000.
In short - No, you cannot store all Pokemon forms and their shiny counterparts in 1500 box spaces at the same time. Sorry!