Hair on a hair system is different from growing hair. Growing hair is supplied with a natural oil - sebum - by sebaceous glands on your scalp. That oil conditions the hair, but also attracts dust and dirt. Hair system hair does not get that coating of sebum, and therefore it doesn't get dirty as quickly as growing hair, and it also doesn't have that natural conditioner. Consequently it doesn't need to be shampooed as often as growing hair. Once a week is enough, unless you have been somewhere very dusty or smoky. Of course, you can shampoo it more often if you want, but it will dry the hair out and knock a bit off the system's lifespan.
Obviously if you are working out every day then you will want to wash your hair afterwards. The point is that you don't need to shampoo it. You can just allow the warm water to run through the hair, through the lace mesh, and onto your scalp. You can shampoo your growing side hair, but just wash your system hair with water. It's not gross. Your system and scalp are clean and fresh and in good condition.
If you have eczema or some medical condition which makes your scalp flake off, that is a different issue. If you need to get access to your scalp every day, I would suggest you either attach the system daily and take it off at night (you could maybe use Got2BGlued) or you could use a light-tack tape at the back and sides of the system. That would allow you to peel the system forward from the back to access your scalp without disturbing the hairline. Then, when you have done whatever scalp treatments you have to do, you could simply roll the system back into position.
