Hi tsgirl,
I have never worked with an independant midwife but I would treat an IM exactly the same as a doula, with professionalism. Like Kelly said they are there to support you and no respectable midwife would have a problem with any support person being there to support a woman regardless of their qualifications. In my eyes there is absolutely no difference between an IM or doula or partner or any other support person. We are all there to help the woman have the birth experience they want. I would have whoever helps make you the most comfortable in your birth suite during your labour.
Hope that helps.