I haven't been on the Lakeshore - but I have on many other routes.
Long distance overnight trains require the bike to be boxed and transported as baggage.
Amtrak has these giant boxes for $10 that no airline or UPS would ever accept.
All you have to do is take off the pedals and turn the handlebars sideways.
Plus there is a $5 bike fee.
That compares pretty well to airline fees that have gone up to $100 for a bike.
Most major Amtrak stations that have checked baggage also have boxes.
Call beforehand to verify and to reserve one, if necessary.
When you say "New York" do you mean the city?
If not, verify if your destination station has baggage service.
Best - J
PS - Amtrak long distance trains are almost always late.