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I would not give the antibiotic to all your flock in their water if that is what you are suggesting as this can lead to antibiotic resistance and is one of the main reasons why antibiotics that can be added to food or water have been withdrawn from purchase over the counter. You would be better to direct dose just the bird that passed the "Lash egg" to make sure she is getting the full dose. Sick birds often don't eat and drink as much as normal and whilst she may not look sick, she is. You would also need to discard all their eggs for at least a couple of weeks if you are dosing the whole flock, so it just doesn't make sense to do that.
I'm not sure what the relevance is of your local feed store expert saying Lash Egg is rare? If you read through this forum, there are plenty of posts about birds with Salpingitis (and quite a few necropsy photos) which is what causes lash eggs. A lot of birds with the infection never manage to pass any lash egg material, but it is still there. It just all becomes impacted inside their oviduct in a large, extremely uncomfortable mass which eventually gets so big it kills them. You may be lucky enough to prevent this infection getting to that stage with proper antibiotic dosing now.
I don’t have a separate pen where I can isolate her from the rest of the flock, so what would be the best way to do this? I have a cage, but she won’t be happy being separated and it’s hot here.