Simon,

To be eligible for a student discount, you must be a student in the US. Additionally you have to sign up for the Student Advantage program, which will cost you $20 US just to join. The Amtrak online site will not complete a student reservation without a Student Advantage number. You can price it without the number, but you can't reserve it without the number. So unless you are a student over here in the US, and I didn't get the impression that you were, you won't be able to take advantage of that discount.

If you do decide that it still makes more sense to book online without the student discount, then yes you can easily pick up your tickets in LA when you reach the station.

But first you will have to reprice things without the student discount to see if it still costs less than buying the railpass.