What they could have had for Alexander is where he studied Earth military culture instead of Klingon. So he doesn't know how to use a Bat'leth, but he knows boxing and how to use a knife (letting him fight in the mess hall, even if he doesn't win). It would be better if he fought and won, thought not in an honorable way (Earth records of old street fighting).
The other idea would have been if he partnered with Starfleet engineering, and learned damage control and repair from them. So during the battle, he goes into a part of the ship that has poisonous atmosphere, to repair something. When he comes out afterwards, they tell him he nearly died in there. Alexander's response would have been, "I can hold my breath for eight minutes. It took me thirty seconds to enter the airlock and reach the site, one minute to remove the cover, four minutes to perform the repair, one minute to put the cover back, and thirty seconds to return to the airlock. I had a minute to spare."
Have his repair turn the tide of battle, and when they pull him out the other Klingons jerk their heads back due to the poisonous atmosphere on his clothes. Alexander sheds as much of his clothes as necessary, punches up ventilation, and breathes fresh air that way.
Have him still fall for the simulation running, and reveal that it was a prank pulled by the prior operator that Alexander fell for because he does not know everything about Klingon consoles yet.
You could even have a Klingon female watch him fight in the mess hall, then during the battle she tells him that they have to deal without the section (that is flooded with poisonous gas). She explains it as "We are Klingons, we do not waste warriors on a fool's errand". Alexander then tells her to watch the console, and when the system is ready, to activate it. He then goes in because, as everyone has been saying, "I am no warrior." He solves the problem, comes back out, and gives his speech.
Maybe Worf chews him out later, but Alexander reminds him of what he learned on the Enterprise, from Riker, where "the ship comes first". What would be cute is if the female Klingon visits him afterwards
At the end, you have the warrior ceremony, where Alexander is welcomed in as a warrior of the House of Martok. He is accepted as his own warrior, not just because Worf is his father.