First, let me state that I'm still a relatively new goat owner in relation to many of the folks here, but I have successfully raised 3 babes to their first breeding season and currently are bottle feeding three more.
That said, this is our method:
We get our babies from a wonderful farm locally that milks about 250 -300 goats daily. So, during kidding season they have a lot of kids to choose from and to take care of! They only feed their babies twice a day and wean about 8 wks.
I get my kids from them anywhere from 4 - 7 days old (they are left on the mom for the colostrom). They said that feeding 3 times a day would be best, to allow their food supply to be broken up into smaller, yet more frequent feedings is more natural and easier on their stomaches. Because we are home most times (we homeschool ) we have decided feeding 3 times a day was best. ( about 6am - 12 noon - 6 pm) They still get about the same volume as feeding twice a day, but broken down into 3 sessions.
As far as weaning, sometimes it depends on the kid. After about 2 weeks or so we start putting grain out and access to good hay and water. Some take to it right away and others takes a bit longer. My new kids right now are about 3-4 wks old and have been chewing hay and green grass for about a week. This week they discovered the grain and are really enjoying it. SO they will probably wean about 8 weeks quite easily. My first 3 (the ones that are about 1yo now) didn't take to weaning until about 10 weeks or so and then much more reluctantly. (of course, I think we enjoyed the bonding part a little too much, so we didn't help! Live and Learn!

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I have also seen a difference between kids that were part of multiple births versus single births. My singles were much more eager to 'grow up' and my multiples needed just a little more time. Of course, they will all continue to bottle feed as long as you continue to give - just like mom in a natural setting. SO. since I am "mom" I begin weaning when they are eating the grain, hay and drinking water successfully and slowly take the bottles away. Ususally this is around 7-8 weeks, maybe a bit longer for some.
Good luck! Bottle feeding is a special time for us in our barn! It gives us a chance to bond with the kid and begin developing our relationship with each animal, us learning their personality and them learning our limits. I am sorry to hear that you will have to get rid of your other goat, but we've been there too. It was best for our herd (we had very 3 bossy ladies that wouldn't let my little ones eat or even rest in the housing) and where they went they found new friends (about 200) and became much more in line with their behavior.
Hope this helps!