Ridgetop
Herd Master
The title probably sounds very cruel and heartless.
It doesn't sound cruel and heartless at all. You are thinking logically. Do you have the time to nurse along this sickly lamb? Do you have the milk, or formula, to feed it? Are you willing to let nature take it's course? In a large flock, this lamb would be a white speck on the grass, one of the casualties a shepherd faces during lambing. It would not stay on the grass for long either, since a predator would take it away for a meal. If you have the time desire, and milk to feed this lamb, then it would be worth it to try. A freezer lamb, even a small runty one, is worth $$$. If you don't have the time, or the milk to feed it, then you need to decide is it worth the effort and cost to try to save it? Since you are able to milk the ewe out, you are inky out the time and effort. Is the attempt worth your time and effort?
Since you have children to help with the feeding, it may be worth it to try to save the lamb. If the lamb doesn't survive, it will be a valuable life lesson for your kids. If it does, you have a freezer lamb for profit, and your kids will have helped to save it. Can your children do the milking or at least some of it?
Just so you know, since this lamb was trapped behind the first born, it may have suffered some brain damage through oxygen loss, etc. If it still cannot stand after a week, that may be what is wrong. Also you don't have to feed it all night long. They are not like human babies. Ewes do not nurse their babies all night long. If this lamb were not so weak, you would only bottle feed him 3-4 times a day. By the time he was a month old you would have him on 2 daily feedings, 1 quart in the am and 1 qt in the pm, with free feed hay starting at a month old. Since he is weak, you can divide up the feedings into smaller, more frequent feedings until he gets stronger. If he still can't stand and nurse on a bottle or his mother by 2 weeks, it will be time to decide on whether or not it will be worth the effort to continue.
If you decide not to try to save this lamb, do not worry about other people judging you over it. Only you can see this lamb's condition. Like I say, he may not be destined to survive if he has brain damage from birth trauma. You have already done a lot to save him, so if he doesn't survive, or gets worse, please move on and know that you made the right choice in letting him go. They are not all able to survive, nature is a harsh mother.