Also you need to spend two hours in Mammoth hot spring although I don't like it very much.
That's 10 hours (4 hours drive, 4 hours in Grand Teton and 2 hours in Mammoth). Since the sunset in Sep. must be before 8pm, you'd better leave Mammoth earlier since the road from Mammoth to Gardiner is a little steep to drive in dark.
For day 2, only one thing to remind: The road between Tower falls to Canyan was closed last year, it may be the same situation this time so you need to check. Otherwise you need to drive 40 miles more from Tower falls to fishing bridge.
If you do want to see animal in Lamar, either after 7:30pm or in the morning (I don't know when). But it's easy to see the bison in Hyden valley (less than 5 miles from Canyan, on the way from Canyan to Lake yellowstone). Also you will find a lot of Elks in Mammoth or Tower falls. But it's difficult to find bear, coyote or moose.
Don't forget to bring binoculars to watch animal, especially bear or coyote.
Have fun in yellowstone.