Any time is the time to bike Hwy 1. Best few weeks I've ever spent. When we did it mid-June a few years ago we almost never felt like we were in danger. Not in the CA sections at least.
Few sections (like Big Sur) are closed so check before going on the road (like I did) and find the surprise.
Can ya get through?
There appears to be a trail for residents across Pfeiffer Canyon, I'm not sure if you can carry a bike through and if non-residents can use it. I'm not sure about the closure near Nacimiento-Fergusson rd. It's hard to find definitive information on what/who can go across and what is open.
Here's a very reliable (and often entertaining) source on road closures in the area: https://bigsurkate.blog/