It sounds like you have a good plan and getting the fecal done was excellent, many people don't bother. I would switch her to cow's milk and doing it gradually is the best plan and it shouldn't take long to switch since she isn't eating much. Once you get the scours sorted out I would just put more milk in her bottle and let her tell you when she is full, within reason of course.
If the scours don't stop then you need to talk to your vet about a bacterial infection such as ecoli or salmonella. I had a set of triplets born in the mud on a nasty night, mom died a two weeks later and all of the kids got scours. Ended up treating them with SpectoGard Scour-Check and it cleared it up. The kids didn't act particularly sick, just not as bouncy as their herd mates.
Please let us know how she does and welcome to BYH.
And there is the catch... hmmm is she lethargic because there is a gut issue? Is she lethargic because she isn't getting enough food?
I understand the first few days of feeding every 4 hours. We raise Nigerian Dwarfs, Miniature Lamanchas & Mini Nubians, and Standrad Lamanchas, Nubians. So we have all different weights and sizes and really go by the individual goat from birth. Not easy with 30 or so kids but usually by a week old we have transitioned them to 3x day except teeny tiny goats that may have come from quads and are only 2lbs.
We have had 4 lb goats take 8 oz right out of the chute so to speak. We are careful in those first week/2 weeks so they don't overeat but our rule of thumb is if they are good, no issues etc then we give them as much as they want at the am feeding which they generally take a lot down. Lunch they may take a bit less... and evening they usually take more like a morning feed. Usually it is 8 am, 3pm, 10 pm feedings.
I can say it is not uncommon that often goat kids are underfed on the bottle. NOT saying this is your case or you are doing this... sounds like you are on top of things. Just often here on BYH a newbie to bottle feeding will lose a kid because they were basically starving. Again, not saying this is your situation but it can be difficult to adjust when this is a new experience. Sounds like you are doing a great job.
As far as whether to slow go the transfer... usually I'd say yes, but because we have had a similar experience years ago we did cold turkey and in our case it worked and had a great outcome... go with your gut...
Meanwhile you may want to consider adding some powder probiotics to the bottle and see if this helps as well.
...and welcome to BYH!
The goats are really cute. They sure are fun aren't they!