Are you exercising as well as controlling your calories? If you are, you will need to increase your intake, as you will be netting far too low. Its fine to do it occasionally, but if you are netting lower than 1000 calories a day then your weightloss programme isn't healthy or sustainable.
I have been using MFP since last June and have lost 3 and a half stone. My calorie goal is also 1200 kcal a day, BUT, if I work out I do try to eat back a good percentage of my extra calories and not net lower than 1000 kcals.
Your diet sounds very healthy, but you will need a decent quantity, as well as quality of food in order to reach a health nutritional intake of all the necessary nutrients. If you start missing out on some of your nutrition, you will find it harder to maintain your eating plan as your body starts craving what its missing, iyswim.
1200 is already a very low intake and no-one would advise eating less than that on a regular basis. The 1200 MFP allocated already includes a really good deficit for weightloss purposes and that's why it allocates more calories for those you burn off through exercise.
I found that after a few months on 1200 kcals I hit a wall for a while. Bumped my kcals up to 1400 a day for a while, by reducing my projected loss to 1lb a week and started losing again straight away. I now have my profile set at 1lb a week (sedentary) and try to eat back most of my exercise calories I'd say about 80% of the time. (I do the 30 Day Shred most days and throw in some Banish Fat Boost Metabolism, walking and using a stationary bike as well.)
You could try adding in a really healthy snack twice a day, something like a small handful of nuts perhaps. That way you are increasing your nutrition as well as helping to reach your calorie goal and you won't even notice it really.
I totally understand what you are saying about it feeling counter-intuitive and odd to eat when you aren't hungry. I was exactly the same at first. It took a while to get my head around it all, but it really helped to go and read some of the sticky threads at the top of the community forums.