I bike commute with a co-worker and sweat copiously (at least I do.) I have never smelled any stinkiness from either one of us afterwards. As others have said, sweating when clean does not seem to produce any bad odor. What I do find objectionable is the feeling of stickiness afterwards. This can be fixed by wiping down in the bathroom.

What I have found out is that eating spicy or garlic-laden food seems to impart a discernible odor to my sweat. While not unpleasant, it does produce a noticeable smell. So maybe cut down on pesto the night before.

OTOH, if you approach Pepe Le Pew levels of stinkiness, maybe that will persuade your manager to be more lenient towards the use of the shower.