After verifying that she was actually transporting like that, and asking for an explanation, unless there's a misunderstanding, I would pull your child.
I guarantee you, if she's disregarding your wishes on this one visible area, there are other areas (not yet visible or not ever visible) in which she is lacking as well.
As a dcp/nanny, I have had some pretty random requests that I've been asked to comply with. For instance, I have a family who wants their children to not be fed grapes. They have a legitimate reason, but it's still kind of random and a bit disruptive at times. Nonetheless, it's what they've requested and therefore I follow it. My dcp don't ask me for many things, because they know I am both reasonable and responsible, so I know that the few things they do ask are important to them.
A dcp should either (a) be prepared to follow the requests of the parent, or (b) explain to them why they can/will not follow them, and then allow the parent to make the decision on the importance of the request vs. the importance of having the child in that particular daycare. If they are not forthcoming about their inability or unwillingness in any given area... that's a problem.
If the dcp is super, awesome, you love her to death, then an additional request of "either install the seat correctly or do not transport" should suffice, and if it does not, you need to find a new dcp. If she gives you attiude when you make your request, then I would venture to say it's not a good environment for your daughter.
JMO. :twocents: