A copy of the driver’s current operator’s license is not a requirement of the Driver’s Qualification File. The contents of the file are listed in 391.51. The only time that a license would be required in the DQ file would be if you accept a copy of the CDL license in lieu of a road test per 391.33. You would keep this copy of the license that you accepted at the time of hire in the DQ file for the duration of employment and for three years after it ceases. The regulations do not require that you include a copy of the most recent license in this instance. You would just keep a copy of the original license accepted to show that the driver held a valid CDL license at the time that the road test would have been given. Refer to 391.51(b)(3) and 391.51(c).
If you want to retain a copy of the current operator license (both CDL and non-CDL) under company policy, the FMCSRs do not prohibit it.