The schedule is currently leave Lafayette 3:35pm on Monday, Sep 3 and arriving Los Angeles at 10:10 am on Wednesday, Sep 5. Now we all know how close reality isn't a match to the printed schedule, so the question is how late can you stand it to be?
Consider going to Kansas City and taking the Southwest Chief.
I believe that you can do that from either Jackson MS or Memphis TN on the City of New Orleans to Carbondale IL, they bus connection to St. Louis, then train from St. Louis to Kansas City. The Southwest Chief is far more reliable. You might be able to make a reasonable bus connection to Jackson MS right out of Montgomery. I believe that in Jackson the bus and Amtrak are together or close, and the time will be around 5:00 pm. In Memphis, the stations are not close, and the time would be about 10:00 pm, and I would strongly recommend a taxi between them, even though it is walkable (about one mile), but unless you are big and tough looking you might not complete the trek.
For this schedule to work, you would be leaving Montgomery on September 2, and catching the City of N. O. on the evening of the second, and the Southwest Chief on the evening of Sept 3. Since the Southwest Chief is scheduled into Los Angeles about 8:15 am, this schedule makes arrival on the 5th a sure thing baring major catastrophe.
The main downside of this would be getting off the City of N O at a little after 3:00 am to get on the connecting bus to St. Louis. The upside is that from Jackson (or Memphis) onward, this is a through Amtrak journey, so they are responsible for your connections. You could also go all the way to Chicago and make a train to train change - CNO-->SWC - there at a reasonable hour. Whether this is more expensive or not, I do not know.
If I am thinking right, Montgomery to Lafayette LA is about 8 hours on the road, so you would be leaving Monty on the morning of the third. If you go the other way around, you would be leaving on the second if you catch the train at either Jackson MS or Memphis. If you chose to drive to KC, that is what? about 16 hour driving from Montgomery. The 3rd is Labor Day Monday, so everybody and his brother will be on the road making long drives. Do you want to be part of that?
George