Although I am not sure if it will absolutely cause your skin to break out-- it may. And it will definitely become a hygiene problem since you mention sweating profusely. What you can do is invest in an environment friendly soap and wash your clothes by hand then hang to dry. That way, you get all that dirt and sweat off of it and you can re-use it later on in the week without feeling dirty and while still helping the environment.
I hope this idea helps.