We are feeding our bottle babies (dairy, mostly lamancha) 6 ounces at a time every 4 to 6 hours for the first couple days, then 8 ounces at a time about every 8 hours for a day or two. By a week old we are giving 12 ounces at a time 3 times a day. By two weeks they are betting 16 ounces 3 times a day.
The frequency is of course going to depend on what you are able to do. If we can feed often we do, but if it is only 4 times a day for the first few days they are ok with that too. We feed 7am 4pm and 8pm because that is what works for us, sorta odd on the hours I know, but they seem to be doing just fine, as did the kids of the last two years on the same schedule.
Other breeds may require different amounts, like less for smaller breeds.
I hope your baby is doing super now, yes they are pigs and will eat nearly all you let them.

If you can give more feedings it will not hurt, it is the amount at a time that would cause the runs.
I don't know why your doe isn't giving more, I wonder if she is lacking something, or if there is something you could give her?? Maybe start a new thread on how to help bring a doe into lactation? I really wish I could help you with the doe, but at least I had an answer for the baby. Hope it helps you.

Oh, and we are just giving them whole cow milk form the store. We bottle feed for at least 12 weeks.