My sister (35) seems to be exhibiting signs of post partum depression and withdrawing from family members. Can anyone who has dealt with the same give me (25F) tips for handling it?

Hello. My sister had her third baby this summer. Within a few days of returning from the hospital, she was irritable and unfriendly. We assumed she needed space since she had a c-section. While she was in the hospital, my parents and several of her siblings took care of her other two children for several days straight.

About a week later, she called up hysterical that she's been crying for days because nobody cares about her baby and nobody is helping her. This seemed unfounded since we spent days taking care of her other children which was the job we were given. She claims we were more active participants when our brother had children, which is not true. She says we went to see his baby more but she has not reached out to anyone or invited anyone over and when we've gone, it's felt like an intrusion. She was mad that our nephew (5) made a fairly innocuous comment about the baby's name and called it "disgusting" that his parents repeated it. She has been extra harsh on our mother and has yelled at her on the phone for several hours about how she plays favorites with her children/grandchildren. My mom has taken the worst heat which seems unfair given that her entire life has been babysitting my sister's other kids for the last few years. My mom is offering to help daily but my sister keeps turning her down. When she suggested PPD, my sister got clearly offended.

She has mentioned crying all the time, loss of appetite, and not eating. She says all of this is because of a bunch of little things my mom did that all added up. There was no obvious inciting incident and as I said, my mom's whole life is her grandchildren.

Can anyone who has been in a similar situation provide me with tips for the most appropriate way of speaking to her about this? Her husband does not seem to be getting involved in any way, nor has he reached out to us. I think I need to speak with her exclusively about the PPD so it does not appear I'm in any way taking sides with my mother over her (I'm not - though I feel bad my mother is being held to a ridiculous standard the rest of us aren't), but in a way it seems like the conversation will need to dance around irrational seeming behavior that's been going on for several weeks. Important things to mention, 1) She had more than one miscarriage, including one before this pregnancy. She was fiercely private about both (only opened up to me once and asked that my mother tell us but none of us talk to her about it) but it has clearly deeply impacted her and 2) She does seem to interpret mental health conversation as an accusation instead of an attempt to help her.

As of right now, my concise plan is to tell her that she's mentioned several symptoms of PPD to me and I'm very worried about her and that she's going through a lot and not confiding in any of us as needed. I will offer to use up some of my vacation days to help her.

I would really appreciate advice on how to approach this thoughtfully

TL;DR: My sister appears to be exhibiting signs of PPD and not to our knowledge acknowledging them. How can I have a delicate conversation with her without offending her, particularly when she's been hypersensitive lately?