First and foremost: nice work! It's good that you are (a) writing this module and (b) looking to release it to CPAN. It is also in pretty good shape for a first go - well done.
I was thinking Business::Stripe::Simple as the module name - is that sensible?
It isn't crazy :-) However, since Stripe can do a great deal more than you are using it for here (according to the other CPAN Stripe modules) maybe putting something about the purpose in the name would be even better. Business::Stripe::Checkout perhaps? Or ...::WebCheckout to indicate the useragent talks to the Stripe servers directly. Of course if you are planning some later features unrelated to checkout then this becomes less useful as a name.
I will have a look through the code later as time permits since this is something I might actually have a use for.