I haven't been to Havre, but have cycled from Portland OR to Whitefish and taken the train back. I've also cycled from there via East Glacier and then southwards. A few quick thoughts:

1. I like the notion of cycling not too far from an Amtrak line. It might even give you some flexibility where you board the train. For example, if you started on the Pacific Coast and headed eastwards you could board in Whitefish, East Glacier or Havre depending on how you got in that first trip. I did something similar ~20 years ago on the East Coast. I bought a ticket that started in Jacksonville FL but also let open options of Savannah and Charleston. I actually got to Savannah and was able to change my Amtrak ticket to board in Savannah (on same train as originally planned), rather than Jacksonville. Try that one with an airline flight

2. I find Amtrak a comfortable way to travel and sometimes fun when the train parallels routes you've otherwise cycled. You do need to occasionally allow some extra time. Over course of ~dozen long haul trips the train has been on time (i.e. within an hour) about 60% of the time. However, when it is late, it has occasionally been rather late. As an example, my trip from Whitefish (in 2011) was delayed ~5 hours. This year there had been considerable flooding in the Dakotas and the train line was even shut there for a while. It was open, but trains would get delayed. Delays accumulate, so you are more likely to have a delayed train near the end of the line that at the start of the route.