Your company's solicitor/attorney should advise on this - as to whether the request is legal and feasible.
This is, fundamentally, about risk: avoiding the possibility of your company being sued even if the vehicle being driven on company errands is not a company vehicle. If it were a company vehicle, then it would/should be maintained up to a certain standard to comply with the risk assessment requirement.
The only way out of this matter is to say to your company: fine, that's what you want, but IF that is really what you want then I won't be able to run errands for you in my vehicle because although I can maintain it up to an acceptable, roadworthy standard (as required by my country's/my State's law) I may not be able to maintain it to the standard that the company insurers might require, and I'm not prepared to subject my own vehicle to your tests. Unless you (a) pay for the tests and (b) pay for any repairs/maintenance to bring it up to "your standard".
If, however, running errands, and having your own vehicle to run those errands, is part of your contract, then you may have no option except to comply with your company's requests.
Hope my convoluted thinking is reasonably clear!
Raindance
