Can we get the matter pairing code saved to the entity? Or a field where we can manually enter it so we have a history of it in an obvious place
There is another FR which basically requests the same thing, on a more generic level.
I suggest you vote there and ask a mod to delete this thread.
In the meantime you can use the workaround in the second post to save the pairing code.
I think the other feature request is the best you might get.
I am not sure the pairring code can be extracted automatically anyway, unless you pair with a Bluetooth adapter directly attached to the HA server.
Pairring with a mobile device will first pair the device with a fabric native to that device and then move the device to the HA fabric, so the pairring code in those cases are not seen by HA.