Short answer--unfortunately, no.
Since your brother is a resident of another State, the transfer must take place from an FFL in the State of Illinois.
Be advised as well--there are other requirements in the State of Illinois to satisfy for the transfer of a handgun.
You also can not initiate the transfer in California, either. You have to start the process in Washington. Find an FFL in Illinois that would be willing to do the transfer.
Don't shortcut this step--it's not worth a trip to Club Fed.
Hiding in plain sight...