Brian I found a workaround for this. It's not super-convenient, but it works. Create a CSV with the contact and add the custom attribute. It'll both import and be editable.

It looks like the import will simply update the existing attribute with the custom property. So, you don't have to populate a CSV with all existing attributes on a contact. You need only provide the email or User ID and the, the custom property, and its value.

This presents as a bug. You can edit existing Custom Properties on a contact. In the absence of editing, you need to delete and recreate the contact, which is a destructive operation.