Just print out whatever receipt you get from Discover Camping.
This short video and trail rules is really waste of time, unless you are backcountry camping for the first time in your life; consider it benefit there will be nobody to force you watch it!
Normally, when visitor center is open, after check-in they give you small plastic card with rubber band you are supposed to hang on your tent overnight. (I have mine now hanging off a nail from kitchen wall, lol). You won't get it now so again just make sure you have Discover Camping receipt. There is ranger station at Whitehorn and Rearguard, in odd case they are there and start checking which is highly unlikely.
Having said all that, for someone coming to Mt. Robson for first time, Visitor Center is part of experience so it is a shame it will be closed. But it is what it is. View is quite nice though from the terrace, you can get the same from parking lot
Something to make you look forward to: