IPad – 1st gen iPad Mini screen replacement

Did you try charging using another USB cable (the one for your iPhone, perhaps) and see if that made a difference? I’d recommend testing this first before replacing attempting to replace any components yourself.

That said, it’s most likely a battery issue.

Here’s a guide on how to replace the first-generation iPad’s battery yourself.

I've never heard of someone being refused AppleCare service at an Apple Retail store in Europe when the product was purchased in the US. Also, all 2012 mini Wi-Fi models are A1432 so you don't need to play a guessing game with LTE bands.

If you read the terms of the warranty, Apple is certainly in their rights to ask you to return the product to the country of sale for service since that is their standard practice. You are basically selecting the laws and customs that cover your warranty and perhaps giving up some rights since European law is different than US law in terms of consumer protection.

In reality, most countries that have Apple Retail presence have good coverage for US products whether you take it to a local authorized center or to Apple. You might have second thoughts if you lived somewhere where the local distributor expects to provide service to products it sold and not those you imported yourself.

As to the quotes - perhaps they were unfamiliar with WiFi bands and that Apple devices usually will reprogram the baseband for WiFi based on location and not point of purchase.