Can you have your midwife (and the OB, hopefully, having reviewed your plan) sign your birth plan and have it 'on file' at the L&D Department? At ours, we fill out paperwork for most everything ahead of time (circ/no circ paperwork etc.) so we don't have to go through insurance information or etc. with the staff at all - just focus on giving birth. Our hospital is relatively rural too.
Is your FP or Pediatrician supportive of the delayed cord clamping? When we decided we wanted to delay with dd2, we talked to our Ped and OB ahead of time, and they were both supportive (no concerns). If OB had not wanted to support it, I would have asked Ped for a letter stating that she was comfortable with our decision, as the physician for our child. If the OB won't bend on it, then a note from another doctor saying that they have no concern, might help.
Who knows, you might go late and get your midwife when she gets back from her trip, too.
I talked to our OB about making sure we delayed cord clamping, and how we wanted to do it, and also "babe immediately to breast" beforehand. He was awesome with our delivery. Delayed cord clamping is not hospital policy here - when a nurse came bustling over with pitocin to do active management of 3rd stage of labor (also something I'd told him I didn't want) - he told her to get it out of the room; it wasn't in my birth plan, I didn't want it, and he hadn't requested it. He told her to throw it away in a different room and that it better not show up on my bill, either.
And he did this calmly and firmly - I didn't even notice, I was so wrapped up with dd2. Dh was the one who told me about it afterwards, very impressed with OB's tone/approach with the nurses.
This was despite the fact that my OB has no opinion on delayed cord clamping and it's obviously hospital policy to clamp immediately so he's not used to delaying.
You may be asking the OB to do things differently than s/he is used to - but s/he does have a midwife in his/her practice, which seems to me to be a sign that hopefully s/he is more open to natural birth and supportive of some of what s/he might consider the "wackier" requests of an NCB mom.
I have heard of people writing on their inner thighs with marker to say, "I refuse episiotomy," so I'd think that you could also write on your inner thighs that you refuse to allow them to cut/clamp the cord.
It might be helpful to tell the OB that you want your dh to be the one who clamps and cuts the cord as well -- don't say just 'cut' because I've heard of OBs who then clamp the cord, and wait for dh to cut it, which defeats the whole delayed cutting purpose.
It's a good idea to do the research and tell the OB how to manage a delayed cord clamping (IIRC, no pitocin for the third stage of labor, whereas pit then is very common in most hospitals). My OB told me that he could respect that unless I started to hemorrhage, then he'd need to immediately clamp the cord and give me pit.
The other thing I did was bring articles about delayed cord clamping - there was one in a Pediatric Nursing magazine that referred to a study with preemies, for instance. OB was fine with it and didn't need those to convince him, but possibly your OB would. Make sure it's peer-reviewed, scientific or medical articles that you are referring to.