Unfortunately getting Stripe fully integrated with recurring invoices would be a huge task.
I think the way to solve this is through some sort of WebHooks implementation on QF, so when an invoice is created from a recurring profile a WebHook is fired and then a 3rd party application can initiate the payment request.
The problem is catering for all the numerous payment processors out there, new ones are popping up all the time.
I know WebHooks have been discussed before and we really want to move this along in the next few months.