How did you get on at the scan?
I had polyhydramios and was told I would be induced at 40 weeks. I went into labour naturally at 39 weeks with very intense contractions, but the baby didn't descend, because the extra fluid prevented it from engaging.
I had an epidural fairly early on (a necessity because the pain was so bad) and laboured away for a day until I had an EMCS due to failure to progress.
There was TONS AND TONS of fluid when my waters broke so be prepared for the gush! also meconium in the water due to long labour, but baby was absolutely fine.
TBH once I was in labour I didn't care particularly how the baby came out, I would really advise keeping an open mind and not being too stuck on one form or another of delivery.
In a straightforward pg/delivery you can afford to be particular over things like not wanting constant monitoring, but with a complication like polyhydramios you need to put your ideals to second place to what's best for the baby. There's such a small window of time for the baby to come out and you would want to know if your baby is in distress which you do instantly with constant monitoring, it's there for a good reason!!!!
Personally I was very reassured by constant monitoring as it showed baby was doing ok in there.
All that matters is that your baby is born safely and polyhydramious does lift you out of low risk and more into high risk territory.
I'm sure you already know but there is a risk of cord prolapse where the cord comes out before the baby due to the waters swilling around, which could affect the baby's oxygen supply adversely, this is why they like the birth to be more controlled.
If you go into labour early get on your hands and knees to prevent the cord from prolapsing and call an ambulance, don't fart around with getting bags and going in a car to hospital in your own time etc.
HTH