Hi there Chris!
Don here from Shopify. :)
It's not usually possible to restrict access to payment methods in this way.
An alternative might be to offer COD as a payment method but include some messaging in your store to let customers know that this is a restricted method only available for certain customers and that any other orders will be rejected.
It might be possible to create a restriction using some custom API coding, however as the checkout area is not customisable on our regular plans for security reasons, your store would need to be on Shopify Plus in order to access and edit this.
If you are looking to move up from the Advanced Shopify ($299) plan and need more freedom and functionality to grow your business you can set up a contact from our Plus team here.
A further alternative might be to create draft orders for your eligible customers so that you can mark them as paid manually when your customer pays via COD. This is not ideal, as you would need to create and send out the order for your customer but I wanted to present as many options as possible here.
All the best!