The limit is c because you have to use cables, radio, or other traditional methods to send the key. The data in the entangled pair would also have to be set at the time the two devices are constructed, so that’s not super useful. It might be useful for single use authentication, but that’s about it.

Don’t think of entanglement as being like one object in two spots. Think of it like identical twins. One twin getting a hair cut does nothing to the other twin’s hair. Similarly, altering a property of one entangled particle does nothing to the other and actually means they are no longer entangled or identical.