Rather than a new "Bitcoin Lightning" payment method they could just use BTCPay server and let the customer choose how to pay but well done for the nicely formulated proposition
I added this suggestion directly to Mullvad github. https://github.com/mullvad/mullvadvpn-app/issues/4435
Mockup below:
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Discussion
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If you look at their on chain checkout UX is clear, and straightforward. I would be curious to see if they choose to pursue the LN request how they build.
I did suggest BTCPayServer in the ticket, and at the same time abstracted the expected lightning checkout requirements so the devs can best choose what works for them.