Per Food in bathroom it is forbidden to eat in the bathroom. Can a lady nurse in the bathroom or is it considered like eating in the bathroom? (sources)
Rav Aharon Yehuda HaLevi Grossman (V'Darashta VChakarta 4:42:7) permitted nursing in a bathroom particularly in a case where doing so would allow for proper modest behavior.
So far, the only source I have seen for not eating in the bathroom is the B'er Heitev (OC 3:2). He extends this law as a kol shekein ("even more so") from talking. I am assuming they are forbidden for the same reason- it is improper. In line with that reasoning is why women did speak in a bathroom- otherwise there could be a breach in modesty.
I would think an infant has a different standard than an adult (do you stop feeding your child if he is expelling waste?), especially considering the following regarding talking (from R' Sheinberg):
While actually expelling waste (nifne): forbidden except for a great need
While on the toilet: permitted for a need (l'tzurach)
Finished expelling waste: permitted
While we cannot apply that directly to eating since that is worse (the kol shekain), it can certainly be used as a snif l'hakel (element to consider toward ruling leniently).