I'm not a dermatologist but I believe after shave balms are intended mostly for facial use only. Of course, many moisturizing-type lotions can (and are) safely applied to other parts of the body, but they are most likely going to enhance the fragrance at it's most optimum potential.
If I were you, I would (as you suggested) switch to using a scented underarm deodorant, if you can find one in a scent you like. Or, just do what I do - buy some empty plastic atomizers (I buy mine from accessoriesforfragrances.com - no I am not affiliated w/ them...) and decant your favorite fragrance into your gym bag. Then you're good to go, whenever you need it.