Two possible approaches: use the game as it recommends you use it. Start with only the information in "The Spokes" section: learn just enough to use 'versus tests', the Beliefs/Instincts mechanics, and the rewards system. Then, add bits as you need them, and as the existing players absorb things.
"Not being vested in the mechanics" is one way to look at it, but you can also look at it this way: the player may be OK with learning the game slowly, in play, over time, through "osmosis" and this approach is actually the one that BWHQ (the game designers) recommend. This approach gets people to internalize the mechanics as they play, and then move on to employ new bits to internalize. However, this approach comes with two huge costs: (a) it demands that the referee, at least, have an excellent command over the game as the referee now has the added burden of teaching the game, and facilitating the learning of the game, (b) and it demands that you play over a long period of time. BW is not a simple game: it's emergent properties are subtle and demand that you spend a long time thinking about how the game works to affect the progress of characters.
If you're not going to be playing for a long time, or don't have a great grip on the game itself, you may be best to avoid the game (or let the player who does own the rules and is keen on the game, act as the referee for a while).
You could start with Mouse Guard instead: it uses a lot of the Burning Wheel concepts, but in a generally simpler fashion. (ed: As Adam points out, MG is not "BW Lite", nor is it really intended to be, nor is it really intended to be a stepping stone to BW; however, I still think it's a generally simpler set of mechanics, and shares a lot of the basic mechanical fundamentals of BW, so I suggest it because it might give one player a lighter ramp to climb, and the other player a game that's similar to the one s/he professes to like a lot.)
If you do decide to proceed with Burning Wheel, then I highly recommend that you get copies of the Monster Burner and the Adventure Burner. MB shows you how to build opposition characters for your play, and the way that it goes about this gives you a lot of insight into how the game designers feels that the game should work. AB gives you a model for learning the game, and provides commentary upon that model, again, giving you insight into how the designers feel the game should work.
In particular, AB includes several adventures that are specifically chosen and illuminated, in a way that you can use to get a group off the ground with BW. The first adventure is at its core a one scene con scenario that could use just the spokes of the rules, but also give you concrete ways to add bits and pieces onto the basic fundamentals of the game.