Domain registration failing

As you say, the most likely reason is that your account currently isn't enabled for paid services. Unless you've already done all these, I suggest:

1. Verify that your payment method is valid and chargeable. That's under AWS Billing Console -> Payment preferences, check that you don't have expired or unverified cards there. Also, in case that possibly matters, make sure your card/bank accepts international charges (the charge will originate from the US, and I assume you're not).
2. Check that you don't have any pending payment verification under AWS Billing Console -> Payments.
3. Trigger some other (small) billable charge, such as running a non-free tier EC2 instance for an hour. Then wait (up to) 24h for the charge to appear in billing, and retry the domain registration after that.
4. As an alternative to 3, or as a potential next step, you could consider upgrading temporarily to Business Support+. That's $29/month, but subscribing to it does in itself trigger account verification. Bonus is that you'll have much faster response times for any associated support cases while subscribed. Once your domain is registered, you can cancel Business Support+.
5. Validate that your support case was indeed sent to the right queue. Hopefully the error message you got gave you the necessary details, but just to be sure: 1/ Should be opened as root, 2/ should be an "Account and Billing" case (not "Technical"), 3/ should be for Service: Domains and Category: Registration issues, 4/ should include something along the lines of "This is my first billable purchase on this account. I believe my account requires spending verification to complete domain registration. The registration API returns an operationId but the operation fails with a generic error. Please verify/unlock my account for paid services."

Also consider https://docs.aws.amazon.com/awsaccountbilling/latest/aboutv2/free-tier-plans.html#:~:text=To%20upgrade%20your%20free%20plan if you've not done that.

Hope this helps (and isn't a complete overlap with what ChatGPT already gave you...)..

A further update...

My regular support case is now 7 days old and still hasn't been allocated. I instead tried to raise the issue as a 'billing' issue rather than a 'domain' issue. That allowed me to get through to the chat service.

Basically as my account was inactive for a period of time then any billing services have been blocked as expected. This is now being forwarded to another team to hopefully resolve in 24-48 hours.