DaMoshe,
I dont know how you can compare the Husband driving to a doula.
I have heard a shiur in which the posek stated that there is no heter for a doula to drive, she may go along with the patient. unless the patient specifies that she really wants THIS doula and she really needs it. Otherwise the doula may not be mechallel shabbos.
On a side note, doulas can get in the way especially if the delivery happens at home the doulas think that they are as qualified as the medics and usually act nastily towards the medics and EMTs. I have experienced this more than once.