While knowing that a dog is another mouth to feed, it's abandoning a living being which (usually) you love and vice versa. Also, having a dog will increase your chances of getting external aid. I personally am much more likely to give money rather than actual objects to homeless on the street with a dog because I know they will spend it on something other than vices given their situation.
Do you have any sort of homeless station or the like where you can get a meal and blankets for the cols? (Or do you live in a volcano where its always hot (looking at you miami))