it sounds really hard Panda especially with another child to look after.
Do you have any friends who could take your older child for a few hours so you can 'sleep when baby sleeps'? sorry your mum is not helping, aside from refusing to take the children, which is obviously up to her entirely, the comments about coping must be infuriating.
Also, could your partner take both kids out for a few hours one weekend morning to give you a chance to rest? even 2-3 hours of sleep without baby present will make a difference I'm sure. Point out to him that he is getting sleep overnight, which you are not.
When you give baby the extra bottle of milk, is it you who feeds him? Could you get your partner to do that feed, and have a few hours sleep yourself then?
My DS was a very very hungry baby too, it was grim those first few months, we tried expressed bottles as well as extra bottles of formula a few times. Funnily enough, they made absolutely feck all difference to his sleep. He just grew and grew and grew though, and was such a chunk by 3 months, he was in 6-9 month clothes. He's 2.7 now and on the 50th centile for everything, so god knows why he wanted to bloat himself so much in the early days 
it will get better soon, honestly :)