Ok, this should now be available. Edit the invite code field after updating and under Parameters > Validation set Ajax Validation to No. You can then implement your own ajax validation rules using CB Query Field. Note server side validation rules will still be applied at time of save. This means you need to be sure the invite code exists, isn't accepted, and isn't expired in your custom validation rules.

One last ask: Is there a way to have this field read only on the registration form? The code is taken from the invitation link anyway and user is not expected to enter it himself.

No, and wouldn't really do anything since the code comes from URL they could change the read only inputs value by changing the URL. It also allows them to navigate to registration manually and supply the code from their email if they like.