Watch the weather.. but I-5 still best
I don't think there is a "safe" route down from BC to Palm Springs. Every route that I can think of has at least a chance of getting snow -- even the coast route through Southern Oregon and Northern California can get snow at times, although it typically doesn't last long.
In my opinion, the best route is down I-5. It's the fastest, widest, and has the highest priority to be kept open and plowed in case of any weather issues. The problem areas from BC are the small area south of Bellingham through the mountains, then southern Oregon through Northern California, and then the Tejon Pass over the Grapevine into LA.
But, if you watch the weather you can make the run in 2 days with some confidence -- I've done the run from LA area to Bellingham, just south of US/ Canada border many times. If necessary, and if you're concerned about conditions in an area, take a later start so that you hit one of these mountain passes later in the day, rather than in the morning. Typically if there's snow they'll work overnight to clear it out, and by waiting to hit the pass at around noonish, the road will have warmed to the warmest it will get that day minimizing any problems. Similarly, planning to allow 3-4 days down instead of 2 longish 10-12 hours days gives you more leeway in planning and driving.
The only variation I might offer since you're heading to Palm Springs, is you can avoid the Tejon pass north of LA, and head east at Bakersfield on California 58 to eventually hook up with I-15 and come down through the east-west San Gabriel/ San Bernadio Mountain Range on the Cajon Pass to San Bernadino. You can still see snow in that area occassionally, but its the high desert and typically only sees snow every few years. Its a smaller road with less services, and you'll be slower on it than coming down the I-5 to the 210 to the I-10 if the weather is acceptable.