Moving a Nitrate sensitive fish


After having successfully treated my sick fish with you I wanted to ask your advice for moving.
I have observed the following sensitivities and requirements for my fish.

1- Roughly 5% of the tank water evaporates every 1-2 days, If I let it evaporate anymore, the fish gets Nitrate shock.
2- If I change more than 5% of the tank water the fish gets Nitrate shock.
3- If I change the water more that 1x a week the fish gets Nitrate shock.
4- My Nitrate water parameters never get above 5ppm in a week time. At this point, I have to do a water change because the GH levels rise to 14-15 drops.

So my water parameters weekly test roughly:
Nitrate 5.0ppm
Nitrite 0ppm
Ammonia 0.25
High PH 8.4
KH 11 drops
GH 12 drops

My water temperature is consistently 66-69 degrees.
We now have algae on the walls of the tank. And I am the only one who touches the fish tank.

We are moving 5 minutes away into a new house, by the end of June.

What is the best way to handle the move for this very sensitive fish?