If public relations was the only issue (the rebranding to Life Valley is fantastic), you'd have a great idea here. Here are a few things to consider:
Although Death Valley has remained the lowest, hottest, driest place on Earth throughout all of its observable history — you are right in recognizing that its weather and climate patterns have been subject to great change.
Timeline:
In the past million years, stages of world-wide cooling caused the average temperature of this region to drop by 10° to 15°F. As a result, heavier snowfall and — here's the kicker — less melting and evaporation occurred allowing for a great ice sheet to spread southward over Canada and the northern USA. In the Southwestern States, glaciation of this magnitude was impossible because it was too warm and dry. Still, many smaller glaciers occupied the high mountains of Nevada, and even the Sierra Nevada ranges were partially buried by an ice sheet 300 miles long.
It was because of this meltwater that there was a succession of inland seas (collectively referred to as Lake Manly) where Death Valley is today. Lake Manly at its height was actually closer to 100 miles long and 600 ft deep 186,000 to 128,000 years ago. It has been hypothesized to have episodically been the terminus for the Amargosa, Owens, and Mojave Rivers.
Several lakes have occupied Death Valley since the close of the Pleistocene Epoch 10,000 years ago, but these younger lakes were quite shallow compared to Lake Manly. Eventually, it gets too hot and eventually output (evaporation) exceeds input ( precipitation) for the system.
In 2005, Death Valley received four times its average annual rainfall (1.5 inches or 38 mm). As it has done before for hundreds of years, the lowest spot in the valley filled with a wide, shallow lake, but the extreme heat and aridity immediately began sucking the ephemeral lake dry.
Numerous small glaciers still remain in hte Sierra Nevada, but the only one left in Nevada is the small ice field on Wheeler Peak — a natural lake forming in this region is unsustainable.
Your suggestion is to create a freshwater reservoir out of Badwater Basin by using a system of pipelines and cascading reservoirs of decreasing salinity in order to help mitigate the effects of impending droughts to the region — in addition to the technological and economical limitations facing your project the natural one of evaporation should prove insurmountable.
As water is pumped into the basin its surface area increases, leading to an ever-more increasing rate of evaporation. You'll need to keep pumping desalinated water at a rate that grows faster than the create of evaporation. In Death Valley, the evaporation rate is 128 inches annually, 77 times the precipitation rate.
Sorry to say this, but it'll never work.