Secondary registration button on home page that takes use to another registration site. This exists, but in this case it should be possible alongside registration through the system, not in lieu of.
Create a payment processor that lets someone register, but asks them to pay/register through an outside application:
User should confirm that they will pay
Payment is set to pending
User can optionally enter external ID as a requirement
Admin can update payment to paid and enter external ID in notes
This could be either/both: