It could be done but not with that piece of crap. Those little under-counter systems are designed for city water.. they won't even handle well water very well..
Reverse osmosis is a polishing process, not a treatment process. While they do make brackish water membranes, you would need to process your waste water before going through the RO system or your membrane life would be very short.
Yup.
What you'd want to do is to take the gray water and run it through a series of sand filters first.. then adjust the pH of the stream to drop out the soaps, then through another sand filter, then neutralize the acid, then through a mechanical filter, and then through a brackish water membrane, and then through either an RO system or a DI system.
Practical? Probably not at your scale..
If you want to save water, use the shower water to flush the toilets and the rest to water your garden or wash your car.