"I'm wondering if it's ok to set up a shower tent in a LTVA? How about on "regular" BLM land with 14 day camping?"
I decided to look into this some more, and as usual the answers are somewhat equivocal. I guess there is: what is allowable, what you can get away with, and what you feel good (or bad) about. Van campers are obviously just tossing their grey water outside, but it's probably not a lot of water in one place either, unless you shower.
At Burning Man they disallow grey water dumping, as they should, given 60-70,000 people all jammed together. However, the video in this post has some interesting solutions. I like the one about putting a large towel in the bottom of your shower pan, and then letting it dry in the sun.
https://www.gonewiththewynns.com/eliminating-graywater-burning-man
In the LTVAs, most RVs are mobbed together, and I for one would hate to see everyone dumping gallons of grey water on the ground under every RV. What a mess. OTOH, YARC Camp is way out away from everyone else, and likewise single individuals may get away with things when camped away from the mobs.
Here is some info from online. Try and figure it out.
http://blog.goodsam.com/25620/
http://www.truckcamperadventure.com/boondocking-and-gray-water-disposa/
https://www.google.com/search?q=dumping+grey+water+on+blm+land
I for one produce very little grey water when camping. I don't wash dishes like most people do, because of the particular way I cook, and I use a doggie dish for personal washing, so about 1/2 a doggie dish of grey water is about all total that I toss out each day. I shower in towns.