Re: branch to collect bath water for grass
the legality may be your biggest issue, depending on state and local codes, reuse of water, and if it needs "treatment before reusing" in most places it is still considered raw sewage,
(the practicality of it and using it I think poses very little health threat (especially since it is your family and there the normal ones coming in contact, if it was a hotel more concern would be in order).,
In rural areas washing machines and some other gray waters have been discharge on to the garden/grass flower bed, and not the "sewage system". for many years,
a problem with storage of gray water is that it does have contaminates in it and will possibly start to stink if not used up as the contaminates are attacked by bacteria.
the only thing I see in your drawing is that the possibly of the trap filling with sediments and possible plugging during the summer months, could consider the possibly of a three way or diverter valve,
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