I wonder if it would be an option to have two inexpensive
bikes, one at each end of the trip ?
How would that help? If you can't comfortably get one bike
on a train, having a second in another place isn't going to
change anything.

=v= It's how the Dutch do it: There's a place at every train
station where you can lock up a bike (or rent a standard Dutch
"Gazelle" bike). In the U.S., a similar approach is being tried
out in the form of "bikestations:"

http://www.bikestation.org/

This works best if you've got a bikestation at *every* train
stop, not just a handful of them, and unfortunately the U.S.
doesn't make biking *or* rail much of a priority.
_Jym_