I'd forget the map holder, it's just extra bulk and complexity that you're better off without in an AR. Fold the map up small enough so you can just hold it in your hand while riding. It takes a little practice but you'll get used to it. No need to put it in a waterproof map case unless it's raining, and even then most AR maps (usually mytopo in my experience) you'll see are waterproof. If you plotted with a sharpie you shouldn't have a problem with your points washing off either.
There are a few benefits to this strategy, the main one is speed. Unless your checkpoint is right on the trail you'll have to spend time removing the map from the bike to carry it off the trail to go find your checkpoint, then spend more time putting it back when you're done. Sure it only takes a few seconds but those seconds add up over the course of the race. It's also tough to refold the map when it's inside a map case and clipped to your bike, probably not something you're going to be able to do on the move, so you'll have to stop for that as well. Holding the map allows you to do all of this on the go, easily.
It's also easier to share the map (as if that would ever happen in an AR). If you need an up close look at the map it's easy to bring it closer to your face and make sure you're not missing anything. The bike mounted holders can be tricky to read on rough terrain as well.
If you have to bikewack you'll be glad you don't have a giant piece of metal on your handbars to get stuck on every branch and briar you pass by. You'll lose a lot of time there too.
I use something like this:
I keep it strapped around my waist. If I need both hands free for some reason, I'll put the map in there. I also keep a locsack in there in case there's a deep river crossing or a downpour that makes me nervous about the map's integrity. I usually don't read the map from inside the case, I use it more as a pocket. I will use it for it's intended purpose on a paddling section when you need both hands to paddle, but that's about it.