When it comes to the work that "nobody wants to do" I think of Simon Sinek's formula: "start with why"
Seems pretty clear to me there is a very strong WHY here - if this role doesn't exist, then invoices don't get sent, then we don't get paid, then the company doesn't exist!
When that's articulated clearly, is it still true that nobody wants to do this? Or are people open to seeing this role as boring but very necessary - and wiling to do it on rotation as [@mention:449833773917801859] suggests?
Some other options that come to mind:
1. Is it possible to outsource any of this work?
2. Can the 'post office' role be broken down into smaller packets that make it easier for people to take on?