You probably are better off getting it rectified, as mentioned on Amtrak's website.
When Making Reservations
US-Canada Border Crossing for Amtrak Canada Trains When making a reservation for services crossing the US-Canadian border, you must provide certain information, including your date of birth, gender, country of citizenship, and detailed information regarding the form of ID that you will carry with you on your trip. Further conditions apply.
Providing inaccurate or incomplete information will subject you to extensive inspection and questioning at the border. Please make sure that you supply us with complete and accurate information.
The information you provide when you make your reservation will be entered into your reservation record and supplied to Customs and Immigration officers in advance to facilitate your clearance. This information will not be used by Amtrak, VIA Rail Canada, or any Amtrak Thruway provider for any other purpose.
The form of identification you carry with you on your trip should be the same piece of ID that you selected and provided details about when you made your reservation.
You must carry the original, valid identification document. Copies, expired or otherwise invalid documents will not be accepted. Each passenger must have his or her own ID.
Passengers traveling from Vancouver, BC should arrive at least one hour prior to departure for border crossing processing. At locations such as Montreal, QC, or Toronto, ON, passengers should arrive at least 30 minutes prior to departure. For those passengers who are not citizens of either the United States or Canada, it is strongly advised to arrive earlier than one hour to complete additional United States Customs and Border Protection processing."