Need Help...PLEASE

My neighbor has been feeding a sweet little black cat for at least 10
years. She put a pass-through door in one of the panels of her garage
and has bedding in there for the cat, although she feeds the cat on
her front porch. The problem is that the cat is looking sickly and
doesn't go in the garage anymore. She just stays out on the porch
even in bad weather. This is Connecticut and the weather is getting
cold and nasty. The cat is eating very little, just dry food. My
neighbor can pet her but then she runs off. She won't let her pick
her up. She set a trap and hasn't fed her for 4 days and the cat has
shown no interest in going in the trap even though she looks hungry.
She wants to trap the cat and have her put down because she is very
skinny and sick looking now and doesn't want her to have to die
outside in the cold and snow. If the trap thing doesn't work, what
else can she do? Is there anything you can put in the food that
would make the cat lethargic enough so that she could be picked up
and put in a carrier? Since she only eats dry food, I realize this
would be difficult. BTW, my neighbor put a few "kibbles" around the
trap to try and get her to go in the trap. No luck! She has tried
holding the front door to her house wide open and the cat won't come
in. My neighbor has a small dog that the cat just loves. When the
dog is outside, the cat brushes up against the dog and follows it
around. Too bad she won't follow the dog into the house. Any help
would be GREATLY appreciated.