If you are willing to leave in the L&C in Onawa, IA, you can go west to Norfolk and connect with the Cowboy Trail in Nebraska. From the end, you can take US-20 west until you connect with US-26 in Douglas, WY. Take US-26 west until you connect up with the TransAm between Lander and Dubois.
The Cowboy Trail is a rail trail (chat or finely crushed gravel surface) that parallels US-20 most of the way so if you do not like the trail, you can hope onto the pavement. The traffic counts on US-20 east of Atkinson, NE are a tad high but US-20 has pretty decent shoulders.
If you need/want to still go through South Dakota, be sure to go through the Custer State Park area and also to ride the Mickelson Trail in western SD. The problem with going through SD is that you will have to dip back south on US-26 to get to the Tetons first as described above. If you come in from the north via Yellowstone first you will have to head south and then double back unless you do not want to head north toward Missoula. If the latter, there is a route that goes from Jackson and reconnects to the TransAm in John Day, Oregon.
Hope you have an enjoyable ride!