“Duplicate Posts” includes roles to the roles list at its “Permissions” page checking this condition:
// role should have at least edit_posts capability if ( !$role->has_cap('edit_posts') ) continue;
So if you wish to see your role there, then add ‘edit_posts’ capability to it. When you turn ON checkbox for a role here, “Duplicate Posts” just adds ‘copy_posts’ capability to it. And DP plugin shows checkbox for a role turned ON if that role has ‘copy_posts’ capability already. That’s it.
With my test example, I do not see my custom ‘Vendor’ role at DP permissions page roles list, but I have access to the duplicate post actions…
I suppose that you have some other condition, which prevents to use DP plugin for your custom role.