Jaron,
Here's my thinking.... in a gym in a nice comfortable one on one sparring session, being fit may never help you if he is just better...
But add a dose of reality. Why don't make it a multiple combatant episode - one where you simply cannot remain still and patiently wait. You have to run.
OR better yet, in your sparring sessions, add a three on one episode several times. You cannot remain still. You must run. As soon as he is stressed this way he will likely flag. Then he will be slower and more tired.
OR, if you are stronger, work on closing techniques. Once you can get in on him you should be able to beat him.
Just both of you do a 200 meter sprint then immediately commence sparring to replicate the crossing of a battlefield.
I am not sure how this works out, but the "meet each other to duel" scenario lacks the realism that really shows how important conditioning is.
Heck, spend the first 15 minutes of your practice grappling. IF you are fit and he isnt' he'll probably be ill from so many wrestling bouts.
Just a few thoughts.
Tim