Susie, water just isn't enough for teen males.
Seriously, the deodorant would be a great investment. Without it, difficult child can claim he stinks because he hasn't got a good enough deodorant. If he uses the deodorant but still stinks, you then have leverage to get him to wash more often.
Try using the "don't waste a wash" argument.
There used to be an ad on Aussie TV, for a brand of family deodorant. It was VERY Australian, the ad concept was "When it finally rains on the red soil plains..." and went on to show a family rejoicing because at last it was raining. They ran about it the rain, danced in it, washed everything they could (including themselves), put on the brand of deodorant (passing it from one person to another), put on clean clothes and all piled into the truck to drive into town to go to a party. And the main point of the ad was, "When it finally rains on the red soil plains, nobody wastes a wash."
From my experience, an adult well past puberty can use those really strong deodorants DAILY to postpone a was to every second or third day, IF they aren't raising a sweat. because it's not just your armpits that sweat. But teens? Who can't sit still? The deodorant will help the BO, but only if they use it daily (maybe twice daily) AND shower at least every second day (probably daily).
But probably actually trying it is the only way of proving your point.
Marg