Can I allow addresses in regions with unusual formats to bypass validation with Address Ninja?

Partly. You can limit most rules to certain countries, but not to specific customers.

Nine rules can be limited by country: Require house numbers, Require apartment/suite numbers, Block P.O. boxes, Validate postal code format, Block DHL pickup points, Block postal codes, Limit characters per field, Block special characters, and Custom rule (REGEX). Open the rule and choose Except these countries to skip the countries you add, or Only these countries to check only the countries you add. Then click Save, or Save and turn on if the rule is off.

Block malformatted DHL (Germany) and meineinkauf.ch format have a fixed country and cannot be limited.

There is no way to skip validation for one customer or one address inside a country.

See How do I limit an Address Ninja rule to certain countries?.