I really think that you should contemplate finding another ped. I am a OB and Peds nurse and find it very concerning that this opinion is held by someone who is caring for newborns. Not only is it recommended that breastfeeding should occur after the first year if mom and baby desire, but 19 pounds, although on the small side, is not unhealthy for a 1 year old. My little girl turned 1 on the 22nd of this month, and weighs 20 lbs. Her ped is not concerned about her weight at all. I find it very disturbing that someone in the medical profession would discourage anyone from breastfeeding, especially someone who is obviously comitted and has done a great job. Since your baby is getting some whole milk already in childcare, I would absolutely be BFing while able, for your baby's health of course. As long as your baby is eating a healthy diet, is active and you and he both want to continue nursing, keep at it.
By the way, about the walking thing....I would not be concerned about that at this point. Has your LO achieved other milestones, like sitting up and crawling (or scooting), if so, don't be concerned.