Never had one with DD - I drove myself to hospital for a check up at 38 weeks and was induced the next day. I'd been pulled over by a police officer for RBT 3 days before, he laughed when I asked if I could stand up to do it because I couldn't get enough puff to blow for as long as I needed to for the test but he never said anything about needing a medical cert to be driving (and it was pretty obvious I was about to pop).

Don't know about the insurance angle though.