This here may agree with the doctor:
www.sciencedirect.com/science/article/abs/pii/S2468867321000754
I would go with papaya leaf extract, but of course that's just me, knowing it works for me. I would think that going for the prednisone because of illness could be bad timing.
I do certainly believe in having options, though, so you need the right help from your provider.
If you suddenly see the bruising, petechia and mouth sores etc., you pretty much know your platelets are low. I have even cancelled a trip to getting a blood test once, because those symptoms told me the platelets were low. That would be the time to have your safety net in place, as you undoubtedly know. I'm not that familiar with sickness bringing on low platelets, but it sounds like you have that experience.
Actually, I should state that a month ago, I did seem to have some kind of quick bronchial infection that did coincide with a surprise plummet.
I am in your boat a little bit, too. After using my dexamethasone that was ordered 6 months earlier, I ran it by the doctor that it would be nice to have a refill, he ignored. I haven't asked again. At least I have the papaya leaf extract as the net, but it would be nice, even so, to have the dexamethasone on hand, too!