I had the exact same thing with my MIL, who actually insisted in being present in the delivery suite while I was being induced.
I was on very good terms with her at the time, and had willingly agreed to have her there to help support the DH as we were told it could be a difficult birth, and as she wanted to be there when her 1st grandson was born, and had been with her other DIL for 2 of her births.
However in practice, it was incredibly stressfull, she had to keep being asked to leave the room while I had x y z done/was examined and I felt incredibly embaressed and self concious at a time when you're trying to be relaxed as possible.
(only in this sense) I was thankfully I had a VERY long labour and she had to leave before I went fully into the 2nd stage - she missed the main event by a few hours.
TBH IMO having anyone there other than your birth partner is a no-no, and most hospitals won't allow visitors until you're in a ward and then only in fixed visiting hours. Would you want her there for an internal exam?
If you or DP aren't happy to tell her you don't want her there until you're ready, then play the card of no visitors/hospital policy.
What I did, in the aftermath when she wanted to come first thing the next day, (along with the 12-15 other members of DH family) was play the visiting hours card and then say that there was no point coming to the hospital at all as we'd be discharged at lunchtime (again I think you're pretty much in, have baby go home at lunchtime in most places) and that it'd be nicer to have them come to us at home that evening.
If she want's to feel special/involved, let her be the first visitor - or in our case just tell her she is and agree to keep it that way
And I may be bucking convention but if you don't want her there, why don't you say so - it then avoids you DP having to deal with any emotional blackmail argument from her over it as at the emd of the day it's your baby and your birth 