Could we feasibly just run desalination stations from the coasts and pipe the water - like oil - to fill up these reservoirs? Setting cost aside for a second, is there any reason not to do something like this?
Anything is possible for enough money but I don't think you understand how much money what you're proposing would cost. In electricity alone desalination costs around 3kwh/m^3. 3.7mwh/acre foot of water. Last year we got 9.2 million acre feet of water in the Colorado River of the allocated 15 million acre feet. So if we say we want to supplement flows with just 2 million acre feet of desalination water, that's 7 Pwh…
An acre foot is 1233 m^3, and at 3kWh/m^3, that's 3.7 MWh/acre foot. Math checks out so far.
Times 2000000, that's 7400000 MWh / 7400 GWh / 7.4 TWh.
I think you're off by a factor of 1000 somewhere in your math. You're looking at ~the output of a large nuclear plant, not 1000 of them.
Or my math is wrong. That's entirely possible too.