I can't speak for California, but I know from other states that you have to get the petition wording approved by the Board of Canvassars before you can legally collect signatures, so starting ahead of the law being signed will be a waste.
What opponents need to do is get various wordings ready and get their petition signature collectors ready to roll. 400,000 in one week in a state the size of California is simply an issue of manpower, it can be done. You just have to have 5x as many people in 5x as many places collecting signatures.
Plus they can contact all registered Republicans, etc. (the GOP will likely give them a list and even mail information encouraging people to sign, which will mean people will be expecting it and will be ready to sign right away). They can get a quick turnaround time on this if they spend the next two months preparing.
The biggest concern I'd have is foot-dragging by the Board of Canvassars, both in approving the wording and approving the signatures. That's where they could run into real problems. The one-week thing is a dirty trick, but those sorts of referendum avoidance tricks have been employed by both parties in the past (the CCW legislation in Michigan was saved from a referendum by a trivial appropriation put into the bill, which left it off limits for a referendum).