1) Not all babies will take a bottle. My breastfed daughter was like that. I tried, ALL kinds of bottles and nipples. She just would not take a bottle of pumped milk from me or anyone else.
But my breastfed son, would take a bottle.
2) If the baby will not take a bottle, he is prone to dehydration. Dehydration in a baby, can come on really fast, and it is dangerous. Know that.
I certainly hope, that she or the Sitter, knows the signs of dehydration.
3) There was a baby that I used to babysit, that was like that and the parents told me that their child ended up in the hospital twice, by that age. For dehydration. And the baby needed an IV.
4) The Mom, needs to try other bottles or nipples, to hopefully find one that her baby will take. Hopefully.
5) HOW is the Sitter, giving the bottle to the baby??? ie: is she holding the baby during feedings, or just propping up the bottle to the baby to drink? A baby that age can't hold the bottle by themselves. A caregiver needs to hold the baby... to feed. And not while lying down... the baby can choke etc.
If the baby, is truly unable to drink from a bottle (some babies do get "nipple confusion"), then it is not the Sitter's fault per say. IF the Sitter is nurturing and really trying... but baby still will not feed... then the Mom needs to try to try something else.
ie: a medicine dropper... can be tried to feed the baby. Or, there are special feeding apparatuses, that can held to feed a baby that is not feeding well. Like a feeding tube etc. Ask a Lactation Specialist... the Mom will need to solve this... or another Sitter will have the same problem.
My friend's baby, had trouble feeding and she went to a Lactation Specialist and the Pediatrician, and she had to use that feeding tube thing... or a medicine dropper.
Or, she needs to get another Sitter.
A baby needs to feed on-demand. And more often per growth-spurts.
You said the baby is already not doing well.
How much longer, can she wait... until her baby is still not doing well?
Can she or her Husband, take time off to find help or to go to a Lactation Specialist to help with the feeding issues?
But again, not all babies will take a bottle.
My daughter, would NOT take any bottle, at all. She would only breastfeed.