Backpacking in Yellowstone

We are planning a 5-day backpacking trip in SW Yellowstone from near Old Faithful to Bechler Ranger Station. Our permit is for 4 people so we are looking for 1 or 2 experienced backpackers with a vehicle since this is a point to point trip.
We will camp near the Shoshone Geyser Basin the first night. The second and fourth night will be near a couple of the few soakable hot springs in the park. The hikes range from 5-8 miles per day with a total of about 33 intermediate to strenuous miles. On reaching Bechler, we’ll drive to Old Faithful to pick up the second vehicle and perhaps have a goodbye drink.