The band aids are meant to keep the area moist if used with neosporin.
Moreover covered cuts healer faster than uncovered cuts.
Also, the constriction of the band aids may prevent the person from habitually opening his mouth too wide even during sleep. Incidentally I would avoid using "tough strip" bandaids. They tend to cause irritations.
What have doctors said about the white area of the cut? Is that the infection? Has your son gotten a skin culture from his dermatologist? The cuts won't heal unless the infection is eliminated. Your son should definitely look in the mirror every day to see if the whitish area begins to disappear. Don't get too aggressive with repeating the soap treatment without following a wait and see strategy.
Btw, what type of soap was he using? The inventor of this treatment suggests not using anti-bacterial soap (probably because too strong), just concentrated soap.