My logical brain would suggest that if a driver works for more than 1 company, he would be under the control, insurance and be paid by the company he is actually driving for at the time. There is no problem with this in my view as I have done this type multi company working myself in the past and know of many friends who do the same. Indeed the comment about agency drivers above is a resonant point.
However, there is always a ‘but’, if you are paying the driver yourself, you are open to some insurance complications as the driver is not under your control and is not driving a vehicle insured by you. You would need to have some sort of drivers agency set up to do this, as seen in the traditional agency sector. It is probably more trouble than it’s worth to go down that path for a one off.
If it were me I’d loan the driver free of charge and get him paid by the other company, after all it might grease the wheels for you with company and future tenders. If your employee, and/or your customer don’t like this arrangement then forget it.
This is just my common sense view of course, no basis in fact.
Steve