Distill (doesn't work well on the go)
Technically you could distill the water, but this takes a long time and uses a lot of fuel. I'm not sure how many ounces of water you can distil with an ounce of some fuel, but I am willing to bet the conversion rate would be too terrible to be practical. And even if you used scavenged wood, you would have to clear enough wood from the area that your impact on the location would be very noticeable.
Adapt emergency survival ideas
What you could do is look at emergency survival information for procuring drinking water at ocean, and similar information for getting water in a desert, and see if you are willing to do anything similar to those.
A lot of "water in desert" guides talk about solar stills where you set something up to collect water condensing on a tarp or plastic sheet. This does not work well for a hike, but you have several thing in your favor: you have a lot of water unlike the desert solar still, and you are planning ahead on carrying gear for this so you can use better resources.
I have a solar oven. I've never been able to get it hot enough to boil water, but I've gotten it hot enough to generate a lot of steam. My solar oven has multiple optional racks and trays and pots and things that go with it, but if I carry only the shell and a tray (the black trays are what catch the energy) it weighs about 2 pounds or less. I have been planning for a while to see if I can attach it to my backpack and have it cook while walking since it doesn't need to be facing perfectly at the sun to reach 150+ degrees F.
I probably could not generate enough drinking water using my solar oven to sustain me indefinitely, but since it weighs less than 1 liter of water I would consider replacing 1 liter of water with the solar oven on a long hike, as it could probably provide more than 1 liter over the entire duration of a long 1-week+ hike. Don't quote me on that since I haven't tried it yet, though I plan to.
I could probably replace the tray, made for cooking on, with something lighter intended only for raising the temperature. But this is only if I was ok losing the food cooking ability. You might also be able to buy or make a solar oven shell lighter than mine. I paid I think $120 or $150 for mine, so an ultralight one would likely cost more than that.
Most people plan ahead to avoid the rain when they go for a hike. However, if you are properly prepared the rain is usually not so bad. If you planned ahead specifically to go when you expect rain, you could bring resources for catching the rain.
If I just worked the math out right in my head, I think a tarp that is at least 10ft by 10ft should be able to collect a gallon of water with less than a millimeter of rain. You should both double-check my math and practice this if you wanted to attempt it on a hike.
If you choose to rely partially on water acquired on the hike, just remember to always make sure you have enough water to last long enough that you can get to a safe location. If you don't have enough water to return to a safe place, don't go any further.
This means you might have to change your plans if you can't acquire enough water while out.
I think there are reverse osmosis devices that are portable. I looked into it a long time ago. I think I came to the conclusion that it was not worth it, similar to your $1000 find.
If you want to see if that answer has changed in recent years you can try to do a search for something like "portable reverse osmosis". I won't bother to do that search and produce the result here, as specific price fluctuations and their current values are off topic in general.
I suppose what I'm getting at is that you should not look at trying to acquire all your water out, and you don't have to haul it all either. For an extended duration you can bring some and try to acquire some. And have a plan B short trip in case acquiring doesn't work as well as you'd hoped.
Whatever you use, you would likely need to practice it some before heading deep into a hike relying on it.
Just remember that for acquiring water out to work well, the methods of procuring water that you carry have to be lighter than the total amount of water they can provide over the entire duration of the trip, otherwise you might as well have just carried it all.