You mean a fee to pay the part time employee/PE? I don't think there would be a "standard" fee. It would be as the market demands and/or the PE demands.

Most states allow engineering advertising as long as it is not false advertising. You also would need to look into your state's requirements for engineering firms in terms of their registration with the state. Some states require principals to be licensed engineers in order to be legal in offering engineering services to the public. Others allow part time employees to be the PE (i.e. Texas).

Check with your state board on that.

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No. The part-time employee is paid for part-time work.
I am speaking of the rest of the time (weeks on end). The firm already has several younger PEs. I am speaking of using the credentials of a more experienced PE to solicit work.

Will the PE be doing any actual engineering work?

You just can't use false advertising.

If the part time (more experienced) PE is always available for the projects you are advertising for then probably OK.
As far as what to pay the older PE to use their name in pursuing work - that would be up to you and them.

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I doubt the value of using the PE designation because most clients have no idea as to the value or necessity for a PE on the job. Emphasize experience as being more meaningful.