More calories burned in "fat burning" zone?

Ignore the fat_burning zone for purposes of losing fat. It's your overall calorie-deficit that matters. If you deplete glycogen stores while you're inspin class, you'll be burning more fat when your out of class. Why would you care when you're burning the fat, as long as you are?

And if you're only consuming 1200 calories, you've already created a deficit unless you're a very small person who's confined to bed all day.

No, we don't burn fewer calories at a higher heart rate. In reality, the work we do is the main determiner of how many calories we burn. The heart rate isn't a measurement of calorie burn; instead, it's a proxy that can be used to (imperfectly) estimate calorie burn.

In general, we burn more total calories doing X minutes of a specific exercise at a higher intensity than doing that same exercise for the same X minutes at a lower intensity, assuming we're at the same fitness level (and environmental temperature, and hydration, and a bunch of other minor stuff). The reason we're burning more calories in the higher intensity case is that we do more of the activity in the same amount of time.

Look at an example: Calories from walking a basically level route can be estimated based on one's body weight and the distance. Walking a mile in 20 minutes (3mph) and walking the same mile in 30 minutes (2mph) will burn very close to the same number of calories. But walking for half an hour at 3mph burns more calories than walking for half an hour at 2mph, because we've covered more distance (a mile and a half, instead of just a mile). Other exercises are loosely similar - same amount of time, higher intensity means more calories burned. Typically, walking at 3mph will also result in a higher HR than walking at 2mph.

The fat burning zone is more about what fuel mix your body is using in the moment, a matter that is of potential concern to endurance athletes, but for those of us simply trying to lose weight and get fitter, fuel mix isn't that important. Total calorie burn is. If we're in a calorie deficit over the course of time, then the deficit will eventually be made up, mostly by burning stored fat. The stored fat may be burned during the exercise, or it may be burned later - doesn't matter for weight loss.

That implies that the best exercise and best intensity is accomplished by allocating how much time we can spend on exercise (while still achieving good overall life balance), then exercising at a challenging but not exhausting intensity we can sustain for that amount of time . . . without becoming so fatigued that we drag through the rest of our day, sapping calorie burn from our daily activity and wiping out some of the exercise benefit.

I can't explain why your FitBit is telling you what it is; perhaps someone who uses one will have a theory. I use a Polar HRM when exercising, mainly for training purposes, and take its calorie estimates with a grain of salt, . . . or maybe several tablespoons of salt.


Edited: Had typed "lower" where I meant "higher"!

For whatever reason, you are getting wonky data from your Fitbit. A few thoughts:
Zones are completely irrelevant for fat loss.
Fuel substrate is completely irrelevant for fat loss.
Calorie deficit is of primary relevance for fat loss.
Getting a 100% accurate estimate of calorie burns is impossible.
Tracking your own data over a couple of months with averaged actual results will be more accurate than your Fitbit. It is just giving an average based on your stats. And it doesn't seem to be capturing your data correctly.

Ann is right, your goal is too aggressive at 1200 calories at 5'10". I'm guessing your have your goal set at 2 or 2.5 lbs per week? I'm a little shorter than you and maintain on 2500 roughly and lose on 2000 at around 1lb per week. You need to be more realistic about your deficit. It is too high at 1200 and 5'10".