There are different approaches, you´ll have to decide which one fits you the best (and which you can sell to your client).
Subcontract: Just pay your additional designer out of your own pocket and invoice to your client the agreed rate for that kind of work. It does not really matter who did it. There should be a profit out of this to pay for the added overhead and responsibility you take. I think this is the normal standard.
Log your managing time and invoice it separately, at an agreed rate.
Design 20h x $50 = $1000
Project managemnt 5h x $60 = $300
This especially if the other designer is invoicing the client himself. The rate does not have to be different, but it could make sense to do it as this kind of service may have different cots and risks attached.
- You can also agree with your client on a fixed or percentual overhead fee like your suggestion.
In any event, you should log your work and reevaluate after some time if your pricing needs to be adjusted - as always when you do new things.