I have found Google bicycle directions good for getting you to the grocery store, but not good for getting you to Iowa. Instead, Google driving directions with the "avoid highways" option usually produces an acceptable bicycling route.
Since you didn't say where in NW Iowa, let me just arbitrarily presume that you want to go to Le Mars. Google driving directions without the avoid highways option shows 620 miles. With the avoid highways option adds only 4 miles and seems to be a reasonable route.
The route piecing together the TA and L&C would add almost 400 miles and would not, in my opinion, be that much better of a route. The ACA service directory, however, would find you a lot of free places to camp.
Two weeks is very generous for the shorter route, and doable but tight on the longer route.
Since you only have two weeks, I'd suggest you take the shorter route so you have more time to smell the roses, see the sights, talk to locals, deal with breakdowns, etc. It depends in part on whether you enjoy the actual riding the best, or whether you enjoy all the other parts of the experience the best.