And YY the way to manage school pick up when they are exhausted by school in this way is to keep it extremely predictable and routine and as short as possible. Go straight home, don't stop anywhere, and don't ask them to make any decisions. Even a nice decision like whether or not to have an ice cream. If she is likely to be hungry then you could ask her at a calmer time at the weekend what (out of a selection of snacks) she wants you to bring to school pick up, and then only ever bring that, but TBH if it always backfires then I just wouldn't bring anything. I would focus on getting home ASAP.
Rather than offering the snack verbally, you could hold one out, or you could make a statement like "The rice cakes are in my bag" or just show her where they are while you're doing the weekend thing. Then if she wants a snack she can get one or ask for one and if she doesn't want one she doesn't have to answer a question about it. Likewise I agree with the poster saying bring a large umbrella for yourself or one each and just put yours up without comment and make hers available in case she wants it.
Make sure you yourself are as regulated and steeled to sort of "beam your zen" at them and not be derailed or feel worried by it. Hydrated, fed, been to the toilet, deep breaths/power pose/whatever works for you to insulate yourself against the bad mood (because it is very catching - I get caught up in DS2's moods very easily and it really doesn't help). If she finds something to shout and complain about anyway, try not to respond in kind, keep your voice steady and low and slow and try either being extra sympathetic (e.g. "Ugh sounds like you've had a bit of a day of it!") or you can go the other way and be jokey in response. Sometimes I get DS to rate the day out of 5. He always rates it 100/0 or 0/100 so it's very clear what his overall impression of the day was.
I do have a rule that afternoon screen time is dependent on safe behaviour on the way home - we are working on not hitting. So any hitting results in losing some portion of screen time in the afternoon. Shouting at the moment is less serious than hitting but if he has got into a loop of screaming and threatening then sometimes I will add that on to interrupt it. It has helped a bit, but I don't think it's helpful to have some big grand punishment that doesn't take small improvements into account - this kind of behaviour is much more of a sign they are struggling than any kind of "entitlement" or "brattiness".
You said yourself it dissipates once she is home, which fits with the restraint collapse type theory.