The cost of managing a project is typically a range of the total cost of the project and is very industry dependent.
For example, your industry may suggest that PM and PM control costs will be 15% to 20% of the total cost of the project. This would suggest that, adding a resource at an assumed 1,800 hours a year, the project would require 270 to 360 hours of PM resources. So you would price the resource for the 1,800 hours and then price some PM resources somewhere between the 270 to 360 hours.
That said, the PM and PM control area is not necessarily linear with the cost of the project. As your project may grow in scope, you do not necessarily need to add the PM resources in an equivalent way.
In your example, it sounds like you are only devoting yourself to the increase of managing an additional resource. If you intend to only decrease your designing time and replacing it with managing time but your overall hours do not increase, then I would suspect you would not "charge" for the PM part of this scope change. Or, if you expect to be consistently working overtime to manage, then you would increase the costs of PM in this change.