Ok I have had a look at your screencast videos.
Firstly, I understand why the Paypal method would fail, so that is fine because the customer has selected the wrong payment method and then tried to enter details into the square fields. You would expect that to fail!
That is this video:
https://www.dropbox.com/s/6mc9y5gvf3...20PM.webm?dl=0
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Now, this video actually highlights the problem:
https://www.dropbox.com/s/i3zc3t5diu...20PM.webm?dl=0
So, a suggestion. To stop this from happening, are you able to force the customer to select the radio button before they can enter their details?
You could disable the SQUARE text entry fields, or even hide them until customers click on the SQUARE radio button. And if customers click the Paypal radio button, the text fields on the square element either remain disabled or hidden from view.
At the side of the SQUARE radio button in BOLD text, or even BOLD RED TEXT, you could have some text saying 'If you wish to pay by credit or debit card, please select this payment method. NOTE: You will be able to enter your payment details once this radio button is checked' - then once the radio button is clicked, the text fields either get enabled OR if hidden, they appear on screen.
At the side of the PAYPAL radio button you could have some text saying 'If you wish to pay by PAYPAL, please select this payment method' - then the SQUARE text fields either remain disabled or hidden from view and the customer can pay by paypal without a problem.
And if the customer happens to try and click confirm order, before they have selected either, it throws up a message something like 'You forgot to select a payment method, please try again'.
This is just an idea by the way! I am trying to help with a suggestion!