Actually last time this also happen when i have my first child. I understand we also need to have our own space if not will go depressed. But i think why not you put your baby at first priority as you do your baby stuff first like pump milk,etc.. when baby is sleeping you try to get some nap also. For housework if really you feel cant manage wait til evening your hubby and mother in law is back ask them to help out abit say you really cannot manage.
As for your mother in law saying that a woman needs to stay at home, i wont agree with her. If your husband understand and let you out to meet your friend, you should go ahead and go out to relax abit. For me last time i always bring my kid to my own mum place and let her look after coz my hubby side no one wans to help. Also i think most guy not so dedicated to look after kiddo so if you really want to have a small catch up with your friends maybe you can ask your mum to help you look after awhile.