I'm having a tough time with this too. I'm ruling out the Steripen because of the funky batteries.

For mild filtration, I'll use a bandana or coffee filters.

For main filtration, I'm planning on using a mixture of chemicals, boiling and filtration.

Chemicals weigh almost nothing, so of course I'll have some.

In the Sierra's, boiling may be the only way to go. I may have a stove, but I want to conserve fuel, so I'm thinking about getting a Ti grill and using wood for melting snow.

As far as filtration, there's the MSR Gravity filter. I think it weighs 13 oz. The issue with that is the filter must be protected from freezing. The upside is it can be used as a water bucket to grab all the water you need at one time. The filter is cleaned by backflushing, so this should be able to provide most of the water for the PCT without needing a new filter.

Basically, my plan is:

If it's above freezing, I'll filter. If it's below freezing, I'll boil. If I'm in a pinch, I'll use chemicals.