Wow, water systems are pretty basic and usually if hot water flows in one spot it flows everywhere it is plumbed. A bypass valve would stop water from going to the WH, the city fill vs water pump supply would be on the supply side. An open or partially open shower of sink valve would dilute the hot with cold. It really seems odd.
Has the vanity and/or shower ever had hot water flowing? The toilet wouldn't be in the equation.
Have you been able to or even tried to contact the manufacturer?
Here is kind of a test to see if water is even flowing past the kitchen, if you have a single valve vanity faucet, turn it on either hot or cold and then slowly change to the other temp. The volume should be virtually the same at either extreme and greater when mixed 50/50.
The other thought is there may be shut off valves for the fixtures not getting any hot.
It would be helpful to see a diagram of your specific plumbing system.
I, like you am just stumped. Keep us updated.