"Bringing a bike" sounds like on a trailer? If not you might consider that the coast will probably keep you out of snow for sure but will take you 3 times as long tho.. you do get to see some pretty country ...unless it is rainiing which it almost alway does in winter...LOL
Hiway 5 thru the siskiyous and shasta area will have snow...but it is such a major highway that unless you are actually in the middle of a snowstorm you should find the road itself clear. You can call ODOT or CALTRANS and find out road conditions. If you are riding, you might have to stop in Ashland Oregon and wait for the snow to be cleared.
I have made this trip in a little VW beetle many many times in winter on hiway 5 and not had an issue with snow. Never been asked to put on chains, etc. I would be surprised if you had any problem....