individual,
It's a lack of uplift, not a lack of evaporation, which is the cause of a lack of precipitation at those latitudes.

Saltpetre, unless you are going to have a few hundred meter diameter holes right down to the artesian basin water, forget about anything but kids stuff in recharging it.

In fact the research shows that most of the water in it comes from PNG rain forest of the Owen Stanley ranges, & the stuff we are pumping now is up to a million years old. Perhaps you should be starting your recharge up there.

People have no idea of the quantities involved. A neighbour turf farmer has 4 of 4Ft diameter high flow axial pumps to recharge his 2 by 5 acre 3 meter deep ring tanks, [turkey nest storage], from floods & rarely gets them full before the flood has passed.

Aidan,
of course there's a lack of uplift because of lack of a big body of water.

Hasbeen look at what Israel has done, with sand
Look at the schemes we have put in place small but done
A couple of posts from a returnee are well worth the read thanks SP
Even the Murrumbidgee re uses water that once went to sea
Looking for reasons why it will not work may be fun
But ignore future developments like cheap power and pumping at you peril

individual A problem with your evaporation scenario.

Inland salt water lake being fed more salt water form the ocean, when the water evaporates the water over a short period of time will become more salty (water evaporates salt does not) you will possibly end up with something like the dead sea.

The amount of evaporation inland as in the Nth of Australia is pretty harsh because of the heat.

I also doubt you would get your desired change from hot to good weather in a short time, long time maybe.

Quote "My guess is that it'd take about three seasons to see a positive effect." You must be joking, start summer -> Autumn -> Winter
there 3 seasons 9 months, NO WAY.

PhilipS,
yes of course that's supposed to be three years. Once the lake is full from rain it can be topped up by controlled doses of seawater. This would assure a gradual desalination over the years. It'd not be necessary to construct a canal. To deliver a few mega litres/day for top-up a pipeline could suffice.
Of course a freshwater feed via a Bradfield scheme would be the best alas, there's always someone who fails to look ahead for more than next week's pay packet.
Flooding Lake Eyre would be a project for generations akin to the snowy scheme albeit only a fraction of the cost & a permanent shelf life.
It can be done easily if only the Greens & other academics are kept away. We're already paying them for doing nothing so staying away would save the cost of interference.