Apple Account suggests "Update Payment Details" while making iCloud+ subscription payment

I don't think there is such as thing as an "Apple Pay Card". Do you mean a card you use through Apple Pay?

Note that what you did simply to get an app that is not available in your normal country was the the origin of this issue. Apple appears to do its best to prevent users from doing that by having the system they do, and trying to get around it can mess things up, as you are finding. Something is inconsistent between the payment country and your account country information so the old payment is not accepted any longer. In general you should try to have things set so that everything is consistent with one country. Your card probably should originate from that country, be billed to you at an address in that country, and of course that is the also the country you should have in your country settings. Change your Apple Account country or region - Apple Support

Note that if you change back to your old country, that app will no longer update.