If you absolutely must kill the sickliest of the fish, then that is your own decision. However, I would NOT advise culling the rest. Curing them will require quite constant water changes because of how small the tank is. Both the size of the tank and the water quality will have a massive effect on their recovery. Now, it may not be possible to get them into something bigger, but you COULD at least gaurantee them clean water. Do you do at LEAST daily water changes of 30% or more? In such a small tank there is over twice the highest recommended maximum of fish, if that made any sense. There is twice the
bio-load there should ever be in such a small tank, and the water quality suffers as a result.
What are the nitrates? Again, it's not hard to remedy a water quality issue in such a small tank, but you need to look out for your fish and their best interests. However, this also means it's your decision to put them down. You have to do whatever is necessary to save them, until it gets to the point where they can no longer stave off the illness and succumb. Have you at least been giving them garlic treated flakes or other immune system bolstering foods?
Sorry I haven't been following any of the progress, but it's been a busy summer for me. Forgive me if I asked anything you've spoken about previously. I wish you the best of luck with the Tetras!