I did a quick experiment with Amtrak web site. I put in Kansas City to Albuquerque for Saturday, August 17th. On the first screen that has the fare ($120), there is also a symbol for the number of bicycle slots available, in this case the train has six slots that can be reserved @ $20 each.
If you try to go to La Junta, CO on the same train on the same day (but without baggage service), then there are no bicycle slots available...as shown in the second screen shot below.
Here is the same train but only going to La Junta
So I think you can get "there" with a bicycle on Amtrak on the Southwest Chief train, but need to pick your stops appropriately. Here is a link to the schedule for that line and you need to use stops with the little suitcases beside them -
https://www.amtrak.com/content/dam/p...ule-073118.pdf
Also note that the PDF schedule above says "Trains 3 and 4: trainside checked bicycle service offered between staffed locations handling checked baggage.
Customers will check in with the station agent, get a claim check/baggage tag for their bike, and hand up to a crew member inside the baggage car."