Welcome to YT!
You mentioned trying "to make it clear she was to go outside", but also that you "hate the idea of her out there in the rain and cold weather". So I'll first say that whatever you choose, consistency is key with potty training
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The breeder I got my girls from "threw old rugs on the floor"... yep, they came home and started peeing on my rugs! It wasn't their fault, they hadn't been taught any differently
One of them tore up pee pads initially too. I ended up putting a potty pad under a rug they already pottied on by the back door so it got their scent, but also so I'd know when they needed to go; but, I was transitioning to go outside.
I really like the idea of the potty pad in the litter box - useful inside and outside in bad weather! If she's already going on the rug on your patio, an option would be to put a potty pad under the rug (or newspaper) in the litter box (so it'll already have her scent on it... then transition to just potty pads or the washable ones - but I too would recommend against any kind of litter.
Also, don't know if anyone here has tried the pee post or training spray, but that might be another option for her to understand where you want her to go.