They did reach out to me informing me that they couldn't replicate the issue (despite when on live chat they confirmed the issue on 3 different devices).
I informed them that I wasn't happy that even after confirming the issue they're trying to tell me it isn't something they can replicate..
We wait with bated breath.
I've had a response from Shopify regarding this.
I recommend you all reach out to them as it seems they are treating this on a case by case basis.
Here's their reply to me just now:
"For the time being there are two work arounds that are suggested. One is to inform affected customers that if they reload the "Information" page of the checkout this will resend the request, and they should be able to checkout as normal.
Or you may wish to disable local pickup until a fix has been put in place.
Regardless, I have added your shop to that affected list and you will receive an update as soon as we have any relevant details or a fix has been pushed."
Just had this email from Shopify:
Hello there, this is Samantha from Shopify's Technical Support Team. You recently contacted us about an issue with customers being unable to checkout on your shop when selecting local pickup. Customers may have noticed "error - payment" in the title text of their browser window. We have made an update to Shopify to improve this issue. Your customers should now be less likely to have the payment step refresh or see "error - payment" when checking out with local pickup.
If you have changed your shop's setup to alleviate this issue please restore your original settings. We appreciate your patience and feedback as we investigated this issue. If you or your customers continue to experience an issue with local pickup or discount codes at checkout, please be sure to let us know. If possible, please provide specific abandoned cart numbers or order numbers involved with the issue. This will help us investigate where the source of the problem may occur.
Hopefully this is now resolved (I'll test this tomorrow as it's 9pm here now).
If anyone is still affected by this issue please reach out to Shopify support because I believe this isn't a blanket fix across Shopify checkout. I believe they're fixing this on a store by store basis.