Some third-year resident physicians would like to moonlight at our clinic, but insurance companies have told us they will not recognize billing by residents. Is there a way around this problem? Can we bill them under me as medical director although I am usually not on premises?

No, incident-to does not apply to residents, and besides, you’d need to be on premises to bill incident-to. Note that you may be able to bill Medicare. The AAMC has a summary of the rules on its 

For private insurers, they call their own shots, although it may still be worth a few more phone calls to try and negotiate a solution, depending on the volume of billing.