I wish I had an idea of what's going on here. I think some further testing, outside of your live environment is in order.
Install a fresh copy of Blesta, run some tests using the included Stripe gateway
Install a fresh copy of Blesta, run some tests using the 3rd party Stripe gateway
Do both work as expected? I think it's important to narrow down the issue to either a problem with the gateway, Blesta itself, or your specific installation. Running a test like this should help narrow it down. If you need a couple trial keys for these test installs send me a PM or open a ticket with us.