Hi Ben,
The Contact lookup field on case entity is based on the value of the customer filed.
The Customer field on the case entity have two types—Account and Contact.
If the Customer value is Account, the contact field can only show related child contacts of the Account.

If the Customer value is Contact, the contact can’t be filled with any value.

And the customer field can be added to the case grid, why you can't?

However, if you insist on making the contact field be updated to the same as the customer field, the OOB contact field on case entity can't achieve it.
Maybe you can create a new 1:N relationship between contact and case entities, and add the new contact lookup field to the case form.
Then you can create a workflow to update the new contact lookup field with value from the customer field, also you can add the new contact lookup field to the case grid.
Regards,
Leah Ju
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