Hi OP, I really feel for you 
I have some things in common with you, I've just had my second child (she's nearly 4 weeks old) and I also have a 3yo. I've also had depression in the past and probably had PND with DC1 although it was never diagnosed and not as severe as yours by the sounds of it, that sounds really tough
DC1 had tongue tie (I got mastitis too) and he also had silent reflux caused by CMPA but we didn't discover that until much later (when we started giving some formula).
Firstly I urge you to get the tongue tie divided ASAP if you haven't yet done so. This made the world of difference for us. The tongue tie division should make breastfeeding much easier. Then you won't have to faff about with expressing (although it can be worth expressing a bit of milk if you want a break and want to avoid giving formula). Also if baby won't take the hypoallergenic formula, you could try mixing it with expressed breast milk, or I've heard that putting a drop or two of vanilla in the formula can help!
Have you cut all dairy out of your diet? There is loads of helpful info and advice on this website including a diary free treats list dilanandme.com/dairyfree/treats-list/
If you're on Facebook there's a helpful group on there too ("Breastfeeding with CMPA and other allergies")
Having a newborn is hard enough but you also have a quadruple whammy of tongue tie, Mastitis, CMPA and reflux - which is really fucking hard. But it will get better. Once the dairy is out of baby's system they'll be much more settled. If you can get the tongue tie and feeding sorted, you shouldn't get mastitis any more.
I don't know about you, but personally I find that sleep deprivation is really damaging for my mental health, and if I can just get a few decent stretches of sleep I feel much better. So if you're the same you should prioritise getting sleep and call in as much help as you can get to make sure you're able to sleep (husband, extended family, postnatal doula or night nanny if you can afford one).
Does your oldest go to nursery or a childminder at all? If not could you afford to send them even if it's just for a couple of days or half-days per week?