I wouldn't stop... heck when not pregnant my normal weight is 112 lbs and Im 5'6. Im normally super thin. Have been my whole life. Im currently pregnant with my third and by the time I am 39 weeks Ill probably be about 160ish.
Just eat ALOT of fat. Healthy fats. Butter, cheese, whole milk etc, olive oil, steak etc. Make sure you are eating 3 meals a day plus snacks. I wouldn't stop breastfeeding. No way.
I would try to get more sleep though. If you are getting up at 430 am you should go to bed earlier. Heck I don't get up till 6-7ish and I am usually in bed by 10 at the latest.