The least amount of mountainous riding would be the Southern Tier. That said the few passes on the ST are not especially easier than those on the Trans America. You do miss the mountains in the east which surprisingly were harder than those in the west on the TA, due to the fact that, while shorter, the climbs were steeper in the east.
Not having ridden it, I can't speak about how the Northern Tier compares, but I suspect that it is similar to the TA. Similarly I can't speak first hand on the L&C, but again would expect it to be more like the TA than the ST.
I am guessing the WE is more like the ST, but again have no firsthand experience.
OK so with that out of the way... I would suggest that you not pick a route on the criteria of which route "will provide the easiest climbing through the mountains". Having done the TA which does spend a great deal of time in the Rockies and the ST which is theoretically easier, I would suggest that the TA is not really any harder. I would also suggest that it is more scenic, more varied in scenery, and gives a better sampling of what the US is all about.
Personally I found the scenery for much of the ST uninspiring, the wildlife less varied and less frequently encountered, and the brown dead vegetation and lack of water kind of depressing. The great folks I met and the good food helped make up for that, but still... Given a choice I'd do the TA again way before I'd do the ST again. The exception would be if for some reason I wanted to go in the winter, in that case the ST wins out.