The payment block shows «Please specify shipping method» while a third-part checkout extension is used

Finally, I reproduced the failure.
Your postal code TN23 5GN and £5 shipping method combination really make the failure reproducible.
But the failure is not caused by the Stripe extension.
To prove it, I have disabled the Stripe extension and chose the Cash On Collection payment method, and got the same failure.
Please see the screenshots below.
I think the failure is caused by your third-party checkout extension (Magestore One Step Checkout), or by the Local Delivery shipping extension, or by some other extension.
I can solve such problems with other extensions on the terms of my paid support service.