You sound like you don't want an induction. And your provider sounds like they aren't open to your choices outside of a narrow paragdim. It may be a good idea to look around for a provider who is willing to listen to you, and forgo the 38 week ultrasound. Or can you ask them to move your due date back a week so that your baby isn't at risk for being too early? You do have the option of simply not showing up after 37 weeks, basicall until transition, but I would not recommend that unless you really have no other VBAC options. Find someone else to pay to advise you.
Inductions do up your risk of rupture, and inductions even in an unscarred mama often lead to a CS. Its not practical to expect a woman to come in and perform labor on demand, especially a full two weeks before your estimated due date.
That said, there are reasons to do an induction that are valid. With my last baby (successful unassisted VBAC), I had labor for four hours. Twice. Moved a couple centimeters each time but mostly it was just two weekends of getting very tired and then waiting. Baby got to a plus one station (her head was entirely into, and even partly out of, my pelvis) for the entire three weeks. Then finally we had the labor that progressed into actual pushing, but the midwife didn't make it in time. I was on the bathroom floor and paniced and wet and cold and just generally it was an empowering but lets-not-repeat-that-if-we-can-type experience. So this baby, we are considering a natural induction sometime when my body is ready, in a calm room with labor going with midwife there I'm very comfortable with this (potential) induction plan.
But your plan above, I would not sign up for. I'd ask for other options, another provider, find a midwife, not go in after 37 weeks, anything. I'd be too nervous about not performing fast enough and the baby not being cooked enough and would they turn off the pitocin when I asked and would all the cervical checks after my waters broke lead to infection and, oy, the list is too long. Too much stress for me! I'm sure you will find the right solution for you mama.