Quick update on my GR86 saga, in case it's helpful for anyone else in the future.

I realized the price gap between picking up in Tokyo vs. Kanagawa or Saitama was pretty substantial. My Shinjuku reservation cost ¥175,560, while the exact same reservation with a pick-up outside of Tokyo cost ¥127,545. That difference is pretty meaningful, at ¥48k (~US$315).

I made a new reservation outside of Tokyo, and our plan is to take the Saikyo line from Shinjuku to Urewa Station to pick it up. It looks like that should cost ¥410 per person, so the savings are excellent overall. Plus, we won't have to drive through the center of Tokyo, for what it's worth. I was kind of looking forward to it, but at the same time, I'm sure it will be more peaceful to not have to deal with it.