What if I only do sleeper on the Chief, but coach on the Limited?
That would bring the lowest possible fare down to $713, $623 for a low bucket roomette on the SW Chief and $90 for a Saver coach fare on the Lake Shore.
Again, the chances of getting a low bucket on the Chief during peak travel season is pretty small, and we are in the middle or late in the window that my experience says it would be likely to appear, if at all, for August travel. So start checking immediately and be flexible as possible on dates.
One technical point, a few months ago the website was having a consistent problem with booking an itinerary of mixed accommodation types (bedroom/roomette, roomette/coach, etc). It didn't throw an error message, the "continue" button just did not respond on the second segment with the different accommodation from the first. I don't know whether they fixed it or not in the interim. You may have to enter the segments separately to get pricing and call to actually book if they haven't.
Well, once you have a reservation, if it qualifies you'll have an option to BidUp. There will be a minimum and maximum bid and you only get one shot at any given accommodation type. Bids close 37 hours before departure on sleeper bids.
However, as I think about it, this trip won't quality. Only single segment cash trips do. Multiple segments, Rail Pass, and AGR point trips don't and this itinerary has two segments. I do not think it is wise to break up a reservation just to have a chance at BidUp, since that endangers the connection guarantee. That is especially fraught when connecting from any western LD at Chicago.