How can I stop verification failed alerts from popping up repeatedly?

I'm not sure what your concern has do to with a Mac. I understand it's affecting a Windows PC, so why do you need an Apple device? What exactly is making your PC laptop unusable?

Are you asking about Apple's two-factor authentication?

Use two-factor authentication - Apple Support

If so you need a trusted device which can be an iPhone, iPad, iPod touch, Apple Watch, or Mac that you've already signed in to using two-factor authentication. It can also be a code sent via text or voice to a trusted phone number. You can also use a security key that can be used for verification when signing in with your Apple ID using two-factor authentication.

As you can see there are plenty of options available to use your Apple ID while you're abroad, but you need something that assures your Apple ID remains secure, and isn't being hijacked by nefarious individuals or equally nefarious automated attempts to defraud you.

If that doesn't address your concern please explain.

It appears you may have been assuming you're contacting Apple on this website. This is not so. Perhaps I should have pointed that out earlier.

To contact Apple regarding this problem or any other, you need to use the Official Apple Support link below.

Ok... for my own edification, can you confirm the popup you're receiving resembles the following:

If you click the blue text "Didn't get..." then the following appears

... and you can use any one of the "trusted phone numbers" you provided in the past.

Are those options not available?

I don't have any Windows PCs so I have no idea how they work.

I understand you do not have one of those trusted devices with you, nor a phone with a trusted phone number that can receive a phone call or text from Apple. Nor do you have a security key in your possession.

If all of that is true it's a show-stopper. You won't be able to use your Apple ID because as far as Apple is concerned, some nefarious bad guy on the other side of the globe is trying to defraud you.

It's all... "for your protection" of course.

Is there someone at home with access to your old iPad? Someone you trust enough with that iPad's numeric passcode, who can read the two-factor digits to you over the phone? That's one possible workaround.

At your convenience, consider adding a "trusted phone number" to your Apple ID. It doesn't have to be an iPhone, nor does it even need to be SMS-capable. It can be anything that can receive a phone call.