The trouble is with (lack of) plural.
In more complicated sentences like this, it helps to simplify the sentence down by removing extraneous information. In this case, the can be simplified down to this:
. . . which character can survive . . . with one another.
Hopefully this should clue you in to what's wrong.
Grammatically, "character" need to be plural.
I always try to figure out which characters can survive in a one-on-one battle with one another.
But even then, the wording sounds strange. A better version would be
I always try to figure out which characters would survive one-on-one battles with one another.
Though of course, the best wording depends on what your son is trying to say.