Everyone should absolutely feel free to initiate a credit card dispute, at any time, and if there’s a ticking clock, should do so sooner than later. The only thing to keep in mind is that, in addition to lost revenue, at the time of dispute merchants are charged an additional fee, which averages $25 to $30, for each claim.
But in a case like this, that can be seen as an argument in favor of launching a dispute, since if their refund policy is haphazard or not being done in order of performance date, disputes would ostensibly spur them on to process things in a more efficient manner - which would presumably help other customers in the same boat who haven’t protested to their CC company. And in any case, if they’re repeatedly urging you to donate instead, even after you’ve made your preference clear, that’s annoying and (unfortunately) concerning, so you’re well within your rights to escalate.