I find the "don't bother"-approach will give me the runs sooner or later especially on summer trips.
I take ordinary kitchen scotch-brites and cut them up with scissors into cubes about the size of ice cubes.
A few drops of washing up liquid goes on each cube, leave over night to dry, then pop them in a zip-loc bag with my cook kit.
In the field, I scrub/wipe the pots with whatever is at hand (grass, twigs..), until pots are "mechanically clean", then use water and scotchbrite suds to really make them clean.
I also make sure to always pour the dirty water out on the ground and not back in the water.
One sud will last you at least a day (three meals) this way. Make sure to squeeze most of the water out of it before putting it back in the bag, otherwise you get a mess, that will also freeze and become unmanagable in winter.
EDIT: Sorry, didn't see the "more natural" part of your original post. If you like, do the "two stage" washing up and use birch for the latter part. It has worked for me in the past. Bring a few suds in a small zip-loc as a backup for when birch isn't available. Weighs nothing, and still saves you the bother of bringing a bottle.