Yes, I do believe that 1200 calories is probably too few calories. That online calculation that you did is probably a lot closer to what you need on a daily basis. I know mine is in the 1700-1750 range, and that's before I compensate for any exercise. The younger, taller, heavier, and/or more active your lifestyle is, the more calories you need just to function.
I think the key to eating more, especially at first, is to eat a little more often. Every 3 hours is already pretty good, but every 2 hours gives you more opportunities to get calories. I'm not saying you should immediately do 200-calorie mini-meals every 2 hours, but maybe continue your current schedule, throwing in another smaller snack here and there. Eventually you'll definitely be able to eat more without feeling sick.