That sounds like a great trek. As far as seeing the Tetons, there isn't any view of them while on those south boundary roads. You can see the west side of them while driving east from Ashton. But when you get into the mountains, the view goes away. In fact, there's not much to look at at all while driving the Flagg Ranch road, as you are in pretty thick forest most of the way.
I suppose you could just drive both cars from Jackson, past the Tetons, enter the park, and leave one car at the end-of-trek trailhead. Then head over to West Yellowstone, then south to Ashton and on to the Behhler RS. I'm not sure, but it seems like it took the same amount of time for me to get from W. Yellowstone to Bechler RS via Ashton as it did from Bechler RS to the main highway on the Flagg Ranch road. And the Flagg Ranch road, like I mentioned, is pretty rough and slow in a lot of places. I was glad to have my 4Runner, as it could be a bit hard on a 2WD vehicle.
If I do hike a bit in that area this summer, it won't be till mid-late August. I'll be near Glacier July 4th, then hope to pop a couple quick trips in GNP before heading up to play in the Canadian NPs for a few weeks. The loose plan is to get back down to Yellowstone and do a couple hikes, and finally hit the Wind Rivers on my way back to Colorado. It's a pretty aggressive plan, and I might tire of being on the road that long. We'll see how it goes.
I hope your trip is a great success. I think it's great that you are doing this for your family. They'll never forget this trip.