Please do not sleep with him, it will not make him stay with you.
I agree with so many others here - he sounds as if he is having an affair even if he isn't and it is merely a crisis re. parenting etc it matter not the result is the same, you have been rejected.
My husband does a total commute of 4 hours a day - he would never ever stay in a hotel. He will go to extraordinary lengths to come home to us even when it means being really late.
You poor love, for someone to tell you that they don't love you so soon after you have had their child is just dreadful for you. I can understand you want reassurance and comfort but sharing a bed although offering a very short term respite from your feelings will, in the end be something that hurts you much more than it heals.
You do need supprt from someone in RL, and although I know it is hard and embarassing to confide in someone else please do remember that his behaviour is not a reflection on you. He is not treating you like this because you have done anything wrong, he is doing it because he is selfish and destructive.
I wouldn't snoop on him, i just cannot see how it helps and you might find out things which hurt you more and do nothing to save your relationship. I think he has been very clear with you about his feelings and although it is hard, you do need to really hear that and take it on. Begging is no good.
Please also try to out some practical measures in place and take some legal advice because making your life financially insecure is not going to help you cope or recover. Please do not worry yet about not having another baby etc, you just do not know what can happen to you. Just look after yourself, try to eat and sleep well (hard I know) because your DD needs you to be strong.
I bet you feel like the bottom has dropped out of your world. My heartfelt best wishes to you and your little one.