having a cheat day isn't a great idea. It tends to set people to end up in binge cycle.
It doesn't sound like you are looking at this as a lifestlye, even though you say you "know it is." Here is what i mean- you said the pizza improved your mood, instead of going with the suggestion to find on plan foods you love. If friends wanted pizza, i would suggest you enjoy the wings with celery and bluecheese. You still go out, have a great meal, be SOCIAL, and are on plan, win, win, win!

Also, if you are eating fewer than 1200 kcal per day and nursing and working out and taking care of a baby and a toddler, no amount of coconut oil or amino acids or any other supplement is going to let your body repair itself properly. Trust me- I'm a trainer and have a Master's in exercise science and nutrition, if that makes any difference to you. While I think the "dangers of undereating" are greatly overstated (terrific thread on this topic in atkins forum right now, you might want to read), the body does need enough protein to repair muscle. Without, there is no "back up" (the body will not use carbs or fats for this).

finally, you are down 3 pounds with a goal weight of 12 pound loss. That is 25% in 2 weeks- celebrate! but not with pizza