'Use in connection with employer's business' doesn't actually cost very much. When I first started working for a civilian employer on a part-time consultancy basis, I checked whether driving to the airport counted as 'use in connection with employer's business' - of course they said it did. By that token using your car to drive to Brize to catch a flight out to the desert sun probably counts the same.... But the insurance company only added a small extra to the premium, so it was worth being honest.
Mileage rates? If you work for any half-decent employer and tell them that it's cheaper than the alternative, they should allow it. Twice this year I've used my own car for 1500 mile trips on the continent rather than leaving it at an airport car park for 2 weeks at a time - and that's also meant I haven't needed a hire car whilst away. Ferry, hotel and mileage all paid for without a murmur - why does the military make such simple things so needlessly complicated?
I hope the 'self drive' MT you get these days is better than the rubbish I remember - such as a Chevette estate with a collapsed driver's seat and a Mini to drive to London with no rear view mirror ("It's on demand, sir....").