When I hear stories like this, I always want to ask: What did you decide when you discussed this BEFORE marriage and before baby? I know that we can't plan for every little thing that happens, but we can discuss and plan according to our basic beliefs and goals. Right now it seems that your immediate need is rest. Maybe you can have your mother-in-law come over and keep the baby while you take a good nap.
Once you're both (somewhat) rested, sit and talk about your future. Decide on your goals, and then figure out the best way for you two to reach them. Understand that right now, this is what it is. This will give you something to hold on to and make you feel better when things get all rattly. The next time she brings up the topic, shut it down--"I am NOT ready to send my baby away overnight; this is currently NOT an option. Let's figure out something else."
Yes, couples need time alone. Couples with 2mo babies understand that this is an uncomfortable stage that their marriage must go through--if they want children--and they learn how to steal time while still having a baby around. You incorporate the baby into your life, and adjust your life to accommodate having a baby in it. It is perfectly okay not to just send your 2mo baby away for an overnight period to an environment that you don't trust.
You will have to change how you spend time together. Instead of snuggling all night, you stay connected by touching each other when you can, every time you pass each other. Full-minute hugs where you just about squeeze the life out of each other and tune out the rest of the world. Quickies when you can, to stay physically and spiritually connected, until you can have sex at your leisure. Talk to each other, to keep the lines of communication open. Find things to laugh at together, even if it's just people-watching.
It sounds to me like the exhaustion is more than your husband anticipated. He's feeling overwhelmed and in over his head, and he's reaching out desperately to his mother, who will listen. Having had a baby before, you might have to be the one to bring the sanity back to your house. MIL just wants to help her son and to spend time with that baby, so she's not going to advocate for the baby NOT staying with her. Create a schedule for her to come over once a week, maybe, and keep baby while you two see a movie or something.
PS. I have to do the math every time I want to say how old I was when my baby was born, and that was just three years ago. AND I'm sort of a math junky. The point is that you're asking for help. I swear, some people will pick apart a potted plant just to prove that the roots aren't in the ground.