Is it obligatory to donate the value of escaped ornamental birds that were caught, raised, and sold, including their offspring, given that searching for their owners would be considered insane due to their low value?

If the lost item (Luqatah) is insignificant – meaning it's something that people of average concern would not bother to look for – then it is permissible to pick it up without publicizing it. As for other lost items, they must be publicized for a full year, and ownership is only acquired after that, similar to a date, a piece of bread, a rag, or anything of negligible value. So, if the bird is cheap, and people of average concern would not bother searching for it if lost, then there is no blame on the one who picks it up for taking it and benefiting from it by raising it or selling it. However, if it is valuable and people of average concern would diligently search for it, then the one who picked it up erred by taking it without publicizing it. He must then donate it as charity if it is still available, or donate its value with the intention of giving it on behalf of its owner.