I am unfamiliar with Cathy's original proposal concerning cancel mod. This is getting a bit off track from the thread, but I do think the judge analogy can go through, if you assume the judge always has full complete knowledge of each player's teams at any given time. However, in the end, I don't think it shows anything significant.
Consider Imprison. If I have Pokemon A with Imprison + Flamethrower fighting Pokemon B with Flamethrower + Thunderbolt, on cart, this is the scenario:
Pokemon B clicks Flamethrower - denied due to Imprison
Pokemon B clicks Thunderbolt - goes through and is locked in
So if I'm telling my moves to this neutral judge, it just means that, if I use Thunderbolt, the judge simply won't let me cancel my moves. If I choose Flamethrower, the judge would say "that won't work, they have Flamethrower", and you get to try again. This
technically doesn't work for cases where you must Struggle, because all of your moves are inaccessible. If you selected Fight at all, the judge would not let you cancel and you'd be forced into Struggle. But Showdown in general doesn't support that generally, so I won't count that against your point (since your moves are immediately available without having to click Fight, it leaks Struggle and as a result opponent movesets early, and is basically a big "hey, switch out!" indicator).
A similar thing happens with trapping Abilities. To give an example, Magnezone has multiple different viable abilities. If a Steel-type attempts to check if the Magnezone is Magnet Pull, to do so, the player risks locking in their switch if the Magnezone really is Analytic. If it is in fact Magnet Pull, it would stop the player from switching. A neutral judge could similarly not let me cancel a switch if I attempted one.
However,
even if Cancel Mod is a cart-possible convenience feature, it still wouldn't provide a positive reason to warrant modding Beat Up. Because if it is cart possible, then that's just it -
it is cart possible, no problems here. But modding Beat Up to not display nicknames is
not cart possible without conjoining another change to it. You
mentioned it yourself in the post - you have to assume every Pokemon has the same nickname, which isn't currently possible with the restriction on duplicate nicknames, and why I offered that as a possible solution.