Carla should be talking to her supervisor about the tasks she needs help with and the hours she needs increased over what you are hired for. Whether your or someone else fills that job is up to the supervisor.
You and Abby must confine your conversations to your tasks for her. You cannot appear to be having side conversations about Carla. Carla should not be negotiating with Abby for more of your time since neither of them hired you, right?
You have X number of hours, and that is supposed to be divided in some formula between Abby and Carla. Stick to that. Within the hours each is allotted, she can decide what you work on. But not interdepartmentally.
You work as the hiring manager assigned you. If you have a conflict on the number of hours or how they are divided, that's who you go to. I realize Abby helped you get the job by advocating for you, but that's handled in companies with some sort of employee referral bonus - it doesn't mean she owns you or that you have a special work obligation to her. If you find that Abby is staying late to get her work done because you are being pulled away, you need to work the hours you're assigned. If she's doing it because there is too much work, she needs to go to her supervisor about re-prioritizing things. If Carla is taking long lunches because you are bailing her out, then you tell the supervisor that you are running out of hours and ask how to prioritize or divide things between you and Carla. Don't say she's going to lunch and leaving on time - that's not your call. But you can say she's unavailable during lunch hours so you have no one to ask for direction.
Do not get in the middle of Abby and Carla, and if Beth is in the mix, then that's getting more crazy. You work UP the system, taking to supervisors. At most, you can say to all 3 women, "Look, let's sit down with Ms. Manager and hammer out exactly how many hours I'm to spend with each of you so we have a clear understanding of what I'm being paid for. It's not my call." If they want to sit down with the Manager, great. If one of them doesn't, then you know who the weak link is. Sit down without her.
If their budget is limited, they are only paying part time hours and that has to be divided. Carla and Beth need to petition for additional hours but you aren't likely to be let go if that's approved anyway (which it sounds like it won't be). You have no obligation to work more hours than you're agreed to. Some people just get all excited about having an assistant, and they find more and more stuff for that assistant to do. If that's sending them into long lunches, the managers can find that out if you just ask for the meeting. It's not your problem to solve, nor is it Abby's.