This really should be a matter which should go through your public liability insurance - your men have been negligent in mishandling her car, so she can claim against your insurers. It's unlikely to make any great difference to your next year's PL premium, it's not like motor insurance.
The questions she's asking are perfectly legitimate - and might well be the sort of questions I'd have asked when I worked in insurance liability claims a few years back. Yes you should tell her who your insurers are (you are obliged to, it's not an option), but yes you should get there first, and put in a report to them, not try to fob her off with a cheap repair. If you'd done that my old car (which was written off by vandals worth only £500) I would be livid. You say that you have accepted the blame for the actions of your employees so you have admitted liability - now let your insurance company handle it, stop arguing. It'll be your insurance company who decides how much the claim is worth and who repairs it - not you. You shouldn't be trying to deal with this yourself - the lady obviously knows what she is doing, whereas you sound like you are trying to get a cheap outcome.