Rehoming a single lamb is going to be difficult. I would just keep him on the bottle but have him going out into the field with the rest of the flock as soon as he is definitely keen to drink from the bottle. He will then come to you for feed but be happy to graze with the flock. If he has a safe space in either the house of a shed for overnight he should be fine. I have had individual bottle lambs living in my goat shed, going out during the day into the field (although with the goats initially) and coming in for the night until they were quite big.
Also as he did not get any colostrum from his dam - he will need Heptavac at three weeks rather than the normal 8 to 10 weeks old, as he will not have any antibodies from ewe colostrum. I would also watch out for any bacterial infections and have a strong AB ready if needed. This lack of antibodies will also make him very vulnerable to all sorts if he is rehomed, especially if then into a bigger flock.
Wrt milk - do you have a dairy goatkeeper nearby? If you can buy raw (and more importantly non-homogenised) goatsmilk for the lamb he will do much better and it should be cheaper for you too.