I'm suspicious of adults who need a "best" friend.
Same here. I've been through something very similar to you as well, OP - in fact, I'm astounded that so many people on here have experience almost identical situations! I haven't been through IVF, but on my part there was significant mental illness, horrible stress with social services, a miscarriage, and a death within my family. Ex friend had lost her dad after he'd been ill for over a year.
Neither of us were there for each other in the ways we'd have liked. Like you, I'm the brave face/soldier on/keep it to yourself type. I just don't ask for help, at the time it was a completely alien concept to lean on friends for support - I've always turned to my family and my DH for that, and always prioritised them in turn.
She, on the other hand, is terminally self absorbed and needy. She always had very high expectations of friends - she's a 'classic' only child and seems to expect the same level of attention and devotion from friends that she always got from her parents. She has had a number of 'best friends', who she relies heavily upon, and expects complete loyalty to the point where she considers you a lesser friend if you don't blank/delete/diassociate yourself from people she's fallen out with/perceived to have wronged her in some way.
It's very hard. You're emotionally and physically drained enough as it is without her melodrama. Losing a parent is devastating - mine are still alive, but my dad has an incurable neurodegenerative disease and we're losing a bit more of him each day. That in itself is all consuming. There's a line which is used a lot on MN, which has always helped me, and that's to never invest more of yourself than you're able to give. I think it's very wise.
Take care, OP. Definitely cool the friendship for now, and I wish I had better advice. Best of luck with the IVF too x