A double elimination tournament can mean you have to lose 2 times before the final rounds to be out of the tournament. Usually, once you get to the semi finals or finals, it is one and done. The reason you do a tournament this way is so the tournament isn't over after just a few rounds. Plus, you got guys coming from all over the world. Who wants to have 1 match, lose, and then it is over. Instead it gives guys the chance to at least have 2 matches. By changing to a single elimination in the final round or rounds it creates even more excitement and you don't have a situation where one guy has to win two times in a row while the other only has to win once.
I was a wrestler in high school, and we had double elimination tournaments. So you keep wrestling until you lost twice, then when you got down to the last 4 guys (which I believe is the same case in the movie), it became 1 and done. A winner takes all final. It can get complicated to know who you match against after each round, but there is a crew that is keeping it organized and telling everyone who goes against who. Sometimes, if you won your first couple matches, you'd get a bye, so you get to skip a round because you haven't lost. When I was a wrestler in high school, if I got to the championship match and had one lose and the other guy was undefeated, I didn't have to beat him twice. It was a winner take all in the last match.
The format in Over The Top and the final match makes perfect sense if you understand the rules of a tournament. Even if in most arm wrestling tournaments today in their double elimination tournament, Hawk would have to best Bull twice, that doesn't mean all double elimination tournaments work that way.
The movie can set any rules and tournament style it wants. Clearly, the rules of the tournament made it one and done at the end. Even if we, as the audience, didn't hear them say it, the outcome tells us it is true. Because the tournament organizers and Bull all recognize Hawk as the winner. It is a 90-minute movie, and they aren't going to explain in detail the tournament format. They tell you double elimination for the story. They need Hawk to lose one match to create drama.
So again, I'm not sure of the confusion. It's a movie, not a documentary. A double elimination tournament where the finial round or rounds become single elimination is not only possible but also used in many tournaments.