For water mucked up by cows I'd recommend carrying a little bit of Alum (which you can find at the spice section of a grocery store). Best to save your filter from the worst of the muck. Add a pinch of alum to a gallon of water, stir it for a few minutes, then wait 30-45 minutes. Figuring out how much to add is the hardest part (more is not better, it can fail to work completely if you do that). Gallon ziplock freeze bags are a great ultralight collapsible bucket for this, and you can try different concentrations of alum in different bags without carrying much extra weight.
Even if you don't carry alum, letting the water settle overnight in ziplocks would help if you're camping near the spring.
So this isn't really on your route but it's good info to know in a pinch - there's a very good spring down off the rim at the Muav saddle from Swamp point. It's a 1 mile hike from the trailhead on a good trail, 800 feet of loss and gain. It's called Queen Anne's spring in this list of water sources:
http://grandcanyonbackcountryguide.com/Grand_Canyon_Water_Sources.xlsUnfortunately that file is mostly about below-the-rim water sources (which is where most of the water is in the GC). Interestingly, it also mentions a spring accessible from Fire Point, which might be less of a detour off of your route. But deer trails and route-finding were also mentioned, so it's not to be taken lightly.