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How about using both? Put the skins on and then put into bishop bag.
May your tarps keep you dry and your backsides remain warm.
2QZQ Hammock Specialties
Hennessy Hammock zipper modifications
Sewn on Tarp doors, Pole Pockets, and Grizz Beaks
Ridgeline and gear organizers, peak storage bags, UQ protectors, 2QZQ tree table!
link to one. I use a home made version. It is a regular stuff sack in which I burned a hole through the bottom using a hot copper pipe (melts the material that way). Nothing fancy, but I like it. Mine is big enough to stuff the hammock and my HHSS in together.
I'm not sure what a Black Bishop Bag is made of, but you could make a custom sized stuff sack out of silnylon. Then, seam seal it. That would solve the issue of setting up in the rain.
Leonard Outdoors Custom Antler Handle Fire Starters
Here's the link to the skins. Bonus - you can buy a hand made hat or ridge line organizer from her site as well.
I've heard good things about all those options given above (and own a pair of Hennessy Snakeskins), but don't forget about....
shower scrubby tarp skins
or...tights/pantyhose tarp skins