Mountain passes, oh my! I guess I really don't know the terrain in the north.. I grew up in Florida where everything is flat. Hope there is a not so mountainous route from BC to Texas. Now I am getting a bit scared.
I think some caution and planning are called for, but there's no need for fear. As others have mentioned, for many areas and most of the year only the highest-elevation parts of mountain highways, through passes, are a weather concern. Assuming you head south from Chilliwack, even in mid-winter the roads down in the valleys are not in snow; it would be different if you headed east through British Columbia and Alberta, so you would only want to do that from late spring to early fall.
One good thing about this snow-in-the-passes effect is that it means that you can just check the weather each day, and if necessary take a break for a day (down in the mild valley) to wait for a bad day up in the pass to be over.
We drove Interstate 15 from Las Vegas to Los Angeles at Christmas many years ago (rental car, no trailer) and most of the trip was desert driving in the sun... but the one mountain pass (perhaps
Cajon Pass - I don't recall exactly) had snow. We just drove through, but many clueless people stopped at the side of the road to wait it out. It was funny seeing these cars sitting in the snow for who knows how long, when it only took a few minutes for us to descend back into the warmth. If they were that unable to drive in a bit of snow, all they had to do was check the weather before they left and pick another day.
One November we drove an RV from Alberta to the Vancouver area. The most direct route includes the Coquihalla Highway, which gets mentioned in this forum occasionally. It is a newer high-elevation road that bypasses the slower route of the older Trans-Canada highway down in the Fraser valley. The forecast was for rain, so we took the Trans-Canada, and sure enough while we stopped overnight in rain near Boston Bar, it snowed up on the Coquihalla only 30 kilometres (20 miles) away - we never saw a snowflake. Pick your route sensibly, pause when appropriate, and there's no problem.