Run two browsers, one logged into each account.
I don't know about the business account problem, but perhaps you could buy
things on the personal account and 'expense' them? Like you would say
buying diesel for company van.
(I wonder if the international bar is Brexit related?)
Re Prime, Amazon put a flyer in a parcel recently saying it was possible to
sign up for an Amazon Business account and it make use of a personal Prime
subscription. Not sure if that works the other way around, bur perhaps sign
up for Prime on the personal account (you could expense it) and share that
to the business account.
Theo