Distill by fuel (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.
Solar distilling portable devices
I have a solar oven that weighs 2 pounds, as much as 1 liter of water. I haven't tested to see how fast it can evaporate water (I really should), but as long as it can provide more than a liter over your trip it's worth carrying 1 liter less and taking this instead. And as an added bonus it cooks food too!
If you don't care about using it for cooking, you can buy or make a device specifically just for solar-distilling water. The first device I looked into on amazon is no longer available (it had bad reviews about how fragile it was), but it said it was 2 pounds and could distill between a ping to a half gallon per day. One review reported a pint per day, another reported getting a liter per day, so they back up the item's claim. Since the 2 pound item was still too fragile for repeated use, a good reliable one might be prohibitively heavy if you are hiking solo.
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.
Take time out for supply runs (they can still be naturally sourced)
Wrt safe distance to water, at least in the Northwestern section of
the NP, I don't think any point is further than maybe 8 km from the
This is an excellent point. Since you're hiking you have access to more options than to simply heading back to the trailhead.
You could chart the distance to all the nearest sources of freshwater and make periodic side-trips to them as needed. It shouldn't take too long to make an 8km detour every 2-4 days.
Or you could even plan your hiking path to intersect these water sources every few days. You would just need to plan a hike ahead of time that snakes around a bit more, but that might be fun.
This is another good "take lots vs get out" compromise. You still need to plan ahead and gather extra water, but not for the entire trip.
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.
Be careful of considering natural freshwater sources as one of these safe places to return to: some water sources aren't available year-round, and you wouldn't want to retreat to one to find it dried up and die of dehydration.
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.