As far as the infant, our suggestion would be to pump (even by hand) a bottle that can be given to the baby if he gets too fussy. This milk does not have to be given by bottle, it can be given by syringe, spoon, or cup. That and a little solid food should get him through the day. One day is unlikely to make a significant difference in his weight gain issues.
However, the fact that you cannot go for 20 minutes without drinking, even when lying down all day, is a different issue. This is the basis for a halachic question. If at all possible, you should discuss your situation with a local rabbi who can get all the information in a timely fashion. If not, drink a lot the day before the fast and start fasting. If it gets to the point that you are feeling very nauseated or light headed despite lying down, then start drinking in& shiurim (measured quantities). The correct procedure is as follows:
You should drink less than one cheekful (maleh logmav) at a time. The average cheekful is 40 ml, but you should measure your own mouth. Before Yom Kippur, fill up your mouth completely with water, then spit it into a cup and measure the quantity. Half of that is your& maleh logmav.
Ideally, you should wait nine minutes between drinks. If necessary, this can be reduced to five minutes or less. If you are sure you will have to drink, it is better to start earlier with shiurim, rather than waiting to become dehydrated and having to drink more than a shiur.