It might affect your business a bit, but I do know of a couple of privately owned fitness studios that have bathrooms and a very small changing area, but not a shower.
I would say, at a minimum, you should have a bathroom that is large enough for people to change clothes without bumping into the toilet, or a separate changing area (much more sanitary than changing in a bathroom stall).
I have a shower at my studio, and it is hardly ever used. One the other hand, I train mainly older adults or people with some sort of limitations, and they are not the traditional ‘gym rats’ who go for a workout before running to work.
Looking at your profile, I would expect higher intensity training with the absolute need for a shower afterwards. Most people will go home but you will lose some business if you cannot provide one at you studio. How much is anybody’s guess but if you providing a really great experience then people will accommodate to some to you.