Neither of my girls has SPD, but two of their friends do. I watched one of the friends struggle in a typical pre-school program -- to the point where the teacher was frustrated, the parents were frustrated, the other kids were frustrated, etc. The little girl was pulled out of the school and her parents found a program for kids with learning differences... and that has made all the difference. O is fantastically happy in her new program and she's progressing at an amazing pace.
So... why the story? I think you're possibly setting yourself up for failure putting your youngest in the same private preschool your others went to. The biggest sign for me is that the teacher isn't on-board. That means she doesn't have the confidence in herself, or the desire, to deal with a child that might require extra attention. And for me, that would be enough red flag to pull him out. If you had a teacher who approached the situation with "lets give it a try. I'm sure we can make Z happy, but if he's not able to control himself or he's disrupting the other kids, we might need you in the class" then I would do it. But the teacher is ALREADY expressing concern. She's predisposed to see any problem your son has a too big or too much. And since this is a private school, there's no requirement that they accommodate your son, or make the process easy on you. Even if your son has an IEP, that doesn't make a difference in private schools - just the public ones.
I would not put my child into a program where I needed to be there (if all the other parents weren't). There are just too many other options where your son will be welcomed.
The IEP process can be long- but ultimately it's the best for children who need different accomodations in the school system. Be proactive and ask the school district to test him immediately to see if he'd qualify next year. That way he'd be in pre-school, but with teachers who want to work with him... not someone who wants to classify him even before he gets there.