I have a lot of years in child care so please consider my comments as advice from their possible point of view.
They do have very strict regulations about what they can and cannot do. Such as time out, that is strictly forbidden. Calling it time away and used as time for kiddo to leave a frustrating situation and calm down.
It's absolutely not for punishment. A child care facility is supposed to use teaching as a method of stopping a behavior. If the center your kiddo is going to wants more time to work with this family they may see what you are missing since it's your kiddo he's decided is his victim. They may see a whole different picture. They know what they've learned from the parents in the past discussions, they know what he's suffered in the past or been traumatized with or if he's just a kid that's mean and lives in a completely loving home.
If you are really put out with all of this make an appointment to sit down with the teachers and the director. Ask them if they can share their plan, they absolutely should not share much because he has the right to privacy just as your little girl does. It should be a general behavior plan style document. Such as when a child hits another child he/she will be removed from the situation, checked for injuries, and have time to calm down. IF a child hits again then.....you get the idea. They should have a plan to work through this sort of situation and if they've done that plan then it is your right to let them know you are done. That you are tired of your daughter getting hurt and you are now holding her teachers completely responsible for each and every single injury from this point forward.
Let them know you expect them to protect your child from all injuries and that allowing this child access to her is the same as letting her climb on a toy that is broken and about to fall apart. They are not protecting her enough. You can see that and they do have some part of the responsibility.
If you have any options for other child care you might consider taking her out. If you are a medium to low income family your child is old enough to go to pre-school at Head Start in the Fall. It's often an all day program from 8am - 2pm. It is sort of a part of the local school district but since it's a 3 year old program it's slightly part of it. If there are kids with emotional issues or behavior problems the school district can send in the psychologist to observe and do some simple evaluations. If your child shows any actions or behaviors that says they may have some disability such as Autism the school district can help find resources for that. Other than that little bit of support they don't do much for a 3 year old program.
I'd do that in a heart beat and then have the head start bus drop kiddo back at the center after school. This way she'd only be in the classroom a couple of hours max with the kiddo she isn't getting along with.
Otherwise you can tell the director it's you or the other kid and see who she picks.