weight loss stopped

First off, it's only been a couple of weeks, so (especially at the weight you're at) it's too soon to say weight loss has stopped. Weight loss slows down the closer you are to a healthy weight, and the less fat you have on your body. That's normal, healthy and to be expected. You should not expect to be able to lose 2 lbs a week at this point (at least not 2lbs of fat). Weight loss is not linear and it's common to go a few weeks without noticeable loss, and then see a sudden drop. Especially true if you are changing around exercise routines or eating different things.

Secondly, and with the above in mind, it's very likely that 1200 calories a day is not enough for you. Do you eat your exercise calories back (or at least 2/3 of them?) I hope so! Assuming you are using MFP's default method and not customising your own calorie goal, I would set it to lose 0.5 lbs a week, and eat at least most of your exercise calories back...

... but bearing in mind that estimates of exercise burn can be inaccurate. How are you calculating exercise calories? If you haven't been eating them back, then it's not relevant at this point (but start eating them back!) But, if you are recording huge burns and eating them all back, there may be an issue with over-estimating your burn which could be cancelling out your calorie deficit. If you're going by the MFP database or exercise machines, then just be aware that they can be very inaccurate.

Thirdly (fourthly?), on your cheat day, are you still weighing/measuring/tracking everything you eat? If not, "really packing it away" could easily be wiping out your calorie deficit. It's fine to have a cheat day, or to eat more on some days than others, but you need to be aware of what you're taking in. At this point, there is less room for error, so make sure you know how much you're consuming on those days. And the other days - you say you accurately record everything - are you weighing all your food with a digital scale? If not, that could also be a problem.

And, if you're already a healthy weight, or close to it, I would recommend focusing less on the scale/bmi and more on body composition - how much fat you have on your body, how you look in the mirror/photos, how your clothes fit etc. Don't get too hung up on numbers.

I am older than you, my weight loss stuck for about a year. I never eat back my exercise calories, but use them as a buffer if needed

My only exercise is walking, not as demanding as yours, just to town, on pavement, about 2 miles

Unfortunately our bodies do what they wish, in whatever order they wish

It really is calories in V calories out

Time is maybe not the answer you wanted, but it is the answer