In your screenshot (below) there is no "Plumb Sink" option, even though it's positioned below a rain barrel installed above it. So it's sourcing water from the water mains, since the water hasn't shut off yet.
But I just tested the same save file I sent, removed all mods (except the Grapeseed map mod - since that prevents the world from loading if removed), and I'm still not getting an option to plumb items to the rain barrels at locations 1 & 3, when the plumbing fixtures are moved around. This is the same issue happening at my base location.
Location 1 (Moved Kitchen Sink - No Plumb Option To Connect Barrels)
Location 3 (Moved Bathroom Sink - No Plumb Option To Connect Barrels)
So the option with the red arrow (below) doesn't pop up, which allows plumbing to the rain barrels from the balcony above, and removes the item from the water mains.
The option at the green arrow is related to the items connected to the water mains (currently), and the item has water that way.
Another test you can do from my original save file:
Remove all mods, load the save, open sandbox settings and turn of the water supply in the world.
Then this should force the plumbing fixutres to look for any nearby rain barrels only.
I'm sure this will result in no option popping up to connect these pluming fixtures to the rain barrels.
I hope this is much clearer.
I have removed mods from the save, loaded into it and tested the layouts, it looks like I can plumb the sinks in any of the layouts you mentioned.
This would likely mean that one of your mods is likely causing the issue, since I cannot replicate it on this same save without mods, unless I misunderstood the issue?