Okay so it's a recirculation system with some filters attached to it interesting but not that novel. The one problem I can see right away especially with a shower and a non fully immersed constantly recirculating pump is the buildup of bacteria at the lowest point.

That's a common problem with any plumbing system your drains are often filled with not so nice fungus and bacteria. Normally you don't care because that water is going into an actual real waste treatment system and then it is recycled out into the environment for further breakdown and filtering before it is effectively reused. Doing this at home it means that you need to make sure your filter can filter at the flow rate you need for an effective shower and treat the water at the same time. The UV light is not going to be enough at an acceptable flow rate for a shower. Additionally when killing microorganisms either with chemicals or UV light those byproducts can be as harmful as the microorganism themselves.

There needs to be an effective way that the lowest point of the system is completely drained and completely dry after use to avoid this kind of bacterial buildup. Even with that it would still require regular chemical cleanings because like all bathrooms regardless of how good your humidity control is mold buildup is a problem. Hidden mold buildup in piping where your recycling water is going to be in even bigger problem.

Not an insurmountable challenge but definitely one that needs to be addressed.