i did a quick check on how driving licenses are issued in CA. i admit that you are now running against the tide, cos even if the USCIS were to give you a temp EAD (cos they give that for one month in most cases). let us hope for the better and that they give a 3 month EAD and it will work out for you.

the reason why i  said the current is running against you is rooted in this reference: http://www.dmv.ca.gov/dl/dl_info.htm#BDLP    . i am also quoting the text that is making your task difficult.

"Limited Term Legal
Presence

All original and pending driver license or identification
cards will expire on the expiration date of
his/her
birth
date and legal presence document (BD/LP)
issued by the Department of Homeland Security (DHS).

Your BD/LP document must have an expiration date 30
or more days from your application date. If
your BD/LP
document has an expiration date less than 30
days,
you will not be able to start a driver
license or identification
card application.

If your BD/LP document has an expiration date of less
than 60 days, you may complete your
licensing requirements
(written/vision/driving tests), but you will
need to
submit another BD/LP document with an
expiration date
of 60 days or more before your driver
license or identification
card is mailed to you.

Before your limited term driver license
expires, the department will mail out an
“extension” letter
with instructions on how to complete your
application
for a driver license.

If your current name no longer matches
the name on your birth date/legal presence
document,
see "True
Full Name" and "How
to Change Your Name" for more information."