Much of your ongoing annoyances sound like customer service problems. Can you shop around for a different specialty pharmacy? In my (limited) experience with CVS [0], they do offer to ship to a local store for pickup. I've no idea if they'd be setup to do this with refrigerated medicine, although it seems better for them than relying on a commercial carrier that will happily leave a missed-delivery tag and let the package sit over a weekend.
Perhaps you could also get your doctor to write a new prescription a week earlier, so you'd have more of a buffer.
[0] Just to be clear, I'm not trying to sing any praises of CVS. I've only dealt with them because they just bought out another specialty pharmacy which I also had a short experience with. That consolidation is presumably another symptom/cause of the ongoing dumpster fire. I'm just trying to offer pragmatism to perhaps better cope within this broken system.
addendum: I've no idea why I'm being downvoted. Is the only goal of sharing personal situations simply to pile on the outrage party? Obviously OP has been at it for a while, and in that sense should have their own situation figured out the best for themselves. But OP is probably overloaded with stress due to the situation and is just thankful it "works" at all, which results in clinging to local optima.
shrug, I see another comment referencing that CVS is mentioned in the article (my "article" says "Access Denied"), so perhaps pointing out that they do actually do their job some times clashes with that.