Start with the drains, because it's the cheapest and easiest. I'm pretty anti-bleach because they are bad for the environment and cause nerve damage in humans. Besides, they disinfect but don't clear clogs, and they just replace the odor you smell with another one. You can google "natural drain cleaners" and will get a recipe for something like baking soda and vinegar and that sort of thing. (Be careful that things can be combined safely. Using ammonia by itself is safe, but with other things it releases harmful fumes. Until you know what you've got down the drain, be cautious.)
Usually in the shower drains there is an accumulation of hair, soap scum, skin cells and so on. If a cleaner doesn't work, you can snake the drain and remove what's down there, then repeat with a deodorizer like vinegar and baking soda. Between the foaming action when certain household products are combined, and the deodorizing factor, you should get rid of things.
The sink is also a possibility - hair and soap scum and skin cells (particularly if men are shaving in that sink, which exfoliates skin cells mixed in with shaving foam) can accumulate. Remedies are the same as shower drains.
Also look around the back and sides of the toilet. Condensation can occur due to the difference in the room temperature and the cold surface of the toilet tank holding water from the cooler basement or underground pipes. If you shut off the water, the toilet can be pulled away from the wall by an amateur but you have to know what you're doing.
And the solution is only temporary for all of the above. Bathrooms are wet, moist places.
It's good that you dealt with the fan. Also, does it vent to the outside, or into the attic? You could have something in the attic itself or in the duct work that is wafting back into the bathroom. There are companies that specialize in cleaning ducts out.
A dead mouse is possible, but that smell does not continue over a long period of time. It's worth looking for though. Look inside the vanity for holes into the wall - they can be very small and little more than cracks. The presence of mouse droppings is another clue.
If the tub is a bathfitter, you mean that it was installed over an older installed tub and tile, right? If it was installed over any moist tile, or if the seal is not perfect and water has seemed in between the 2 layers, you could have mold. There are many kinds of mold and mildew, many colors (brown, yellow, green, white), and many smells.
There are home tests for mold you can buy in any large hardware store. As for who to call, you can start with a plumber or a contractor who specializes in bathrooms. They deal with mold all the time.
Mold can be a problem for people with allergies and asthma, and because it appears in moist areas, it can eventually rot through the sheetrock that's behind the tub surround. Having it all torn out is definitely an expense but start with the drain issue and see where that leads you.