There is an armor set for climbing, each piece giving you a climbing speed boost, and with all three equipped you get climbing jump stamina boost. Climb a few steps, jump, slip, repeat. Like I said I haven't found a mountain or a tower that I couldn't climb when it's raining with enough stamina, this trick, and the gear. Plus you can replenish and expand stamina with food and elixirs any time mid-climb (which I almost never need to do, but it's an easy option if you need it).
The complaints about rain are overblown. I like that the game's weather is dynamic, random, and challenges you. Stop being wussies.
It's a dumb mechanic, tbh. I just teleport to a stable and sleep until morning if I see rain is gonna happen for most of the day. Faster that way.