Hi Petra
Yes, I'd go with the steam cleaner, particularly if it has a vacuum unit on it. Alternatively, moulded walls are easy to clean with a large window squeegee and, because there's no residue, it will look cleaner once its done (rather than having smears from cloths etc). Because of the nature of the environment you wouldn't need to use anything else because you won't get a soil build up.
The floor is obviously the hardest part to clean and if they are non-slip tiles you may get a build up of body fats which are best cleaned with a mild acid based product. Obviously, rinse thoroughly (which shouldn't be a problem in a steam room).
Any other awkward corners and fittings I would have thought could be done with even the smallest steam cleaner.
'Heavy traffic remover' sounds like it may be an alkaline degreaser which would be no good for limescale or bodyfat removal - does the floor look dull by any chance?
Regards
Mike