I liked this.  It's amusing, and you handle Aeka and Ryouko
well, balancing the rivalry, tolerance, and friendship.  Sasami
is perhaps a trifle too smart, or too devious, but acceptably so.
Tenchi is his usual nonentity self.

Washu, though, seems to owe more to the TV Washu ("greatest mad
scientist in the universe") than the Washu of the second OAV
series, the competent mother and fairly responsible leader of
the whole group.  I can't see the OAV Washu losing a created
universe at all, and have trouble with the Jusenkyou prank
(at least, without Tenchi's prior agreement).

Your description of Washu developing the prank was good, however:
I really had the sense that you'd thought through the design
problems.

"If I read Tenchi right, just now... hmmm. My little present may
actually do some good. Could break this little amorous stalemate."

For OAV continuity, there is no amorous stalemate, except perhaps
in Tenchi's mind.  As you cite, Aeka and Sasami are from a culture
which allows co-wives (the fanfic sort-of-convention that there's
a Jurai limit of two has no basis that I know of), and Youshou,
Funaho, and Misaki seem agreed on at least four of the girls for
Tenchi at the end of episode 13.

Tenchi, male this time, sleepily wandered out of his room and entered
the bathroom down the hall. After taking care of certain morning
necessities, he groggily filled the wash pail with cold water and then
sat down on the stool near the furo, facing the door. He scrubbed
himself down for a few minutes, and he then dumped the pail over his
head, rinsing the suds off.

The image I get is that the toilet, sink, and furo are in the same
room, as would be the case in the west (w/ "furo" substituted for
"shower" or "bathtub").  This is almost never done in Japan, however,
and I've never seen any sign in the Tenchi OAV's that Tenchi's house
is any different.  The practice is for toilets to be in little closets
by themselves, completely separate from washing facilities.

Even if you did not intend for "certain morning necessities" to involve
a toilet, there's no reason to believe that Tenchi's house would have
a furo on an upper floor, and every reason to believe otherwise.  It's
normally on the ground floor of Japanese houses, like the Tendo's house.

This may seem like nitpicking, but attention to details such as this is
necessary for verisimilitude, for at least some of your readers.  On the
plus side, this is the only such item I picked up; the rest of your
story, which included many cultural details, seemed quite authentic to me.