You have a problem which is not uncommon for babies who did not get enough bottle practice, period.
I want to share a story with you which, I hope, will stop the blame game between you and the sitter because if it continues, you could very well lose it and lose a good sitter as well.
A few years ago, my husband's coworker was desperate to get some time in at work and dropped off her daughter to me. The daughter had been 'mostly' breastfed and wasn't taking a bottle with regularity because, like you, mom wanted to keep her supply up.
I spent four excruciating hours with a baby who simply would NOT take the bottle her mom left her. I called the mom and asked if, after an hour of unsuccessful attempts, I could modify the nipple so more milk could come out. (Bear in mind, I came to that moment having helped raise up nearly an entire generation of babies-- without having earlier encountered this problem.) This little girl refused to take the bottle and I finally called the mom to come pick up her kid. Imagine my surprise when she seemed upset with me, nursed her baby and took off in a huff. While, at the time, I was somewhat floored that the mom had such an attitude with me, later she admitted that her baby hadn't been taking bottles with any real success to begin with and that she'd hoped that maybe the baby would with me because she really had wanted to work. I got stuck in a situation which that mother created entirely on her own.
This has nothing to do with these babies 'trusting' us- hungry babies WILL eat. I don't think I've ever seen a hungry child turn down food from anyone because of the person presenting it. Ever. This has everything to do with the fact that the baby didn't get enough regular practice with the bottle, and that is something babysitters cannot control. She can't go back in time and undo all the times you did nurse when you and your husband should have had the bottle in play. We started our son on a bottle at about 9 or 10 weeks because I didn't want him to only be able to nurse; and you know, it took us a few days of me leaving the house and letting my husband feed our son so that they were able to get their own rhythm. For a breastfed baby, bottle-feeding has to be a skill which gets regular use as, even with a 'breast' nipple, it's a different level of nursing/suckling action.
So, stop with the blame game. My suggestion at this point would be to try a cup. As for not being able to hold a baby constantly-- that's reasonable. I carried babies a LOT and needed to go to physical therapy for my back. You have two other children who need her attention as well; at some point, we do need to put the baby down and tend to other things. It's a reasonable comment. It's hard, because I understand how much we love our babies... mine slept on ME a lot during his infancy, but I can also tell you that when I was a nanny, I did ask parents to let me start helping their child learn to sleep *off* mom and dad, because when we have caregivers, we have to remember that they need to be able to do their jobs with all the kids in their care, including our older ones.
So, back up from "baby doesn't trust you", start using bottles or cups as often as possible, and try to see the reality of your care situation. Maybe you do have a higher-needs kid and may need to hire a nanny who will be willing to pack her around? Maybe you want to let go of some of the expectations about how much the sitter is supposed to hold an endlessly-crying baby. But the fact that this has gone on for six months... I don't know what to tell you. You may have developed too many bad feelings between each other. Personally, when I was working as a nanny, I wouldn't have stayed that long with someone who was blaming me and 'trust issues' for their kid not taking a bottle. It doesn't ring true for me.