Is it possible to add a delivery option to a document package that only certain users can use/see as an option?

No, currently, it is not possible to add delivery options on DDP's for certain users or profiles. 

You can limit availability to Non-Mass, Mass Only, or hide the delivery option altogether. This setting is found on the "Limit Availability" field on the delivery option. 

A workaround would be to create different DDP's with the specific profiles or roles on the set security section with the desired delivery options.