Welcome to the forum and you have got some great suggestions here. I too think taking her out a lot and walking her on a leash (with treats, if needed) might do the trick. I get it that it's convenient to let a dog out into the garden, but for dogs, going potty is also a social thing (at least for male dogs, but I guess also for females) and they miss out on that if they only pee in the back yard. And if she is going in the crate (and previously in your mum's house) because that is the only place she can mark, I guess taking her out and letting her expand her territory, so to speak, might be good for her and for your house.
Although I like the idea of a doggie door per se, I would only let a dog use it if someone was at home, for the reasons @
sisters3 stated (and I would also always oversee my dog outside as I would be afraid someone might want to take him).
I would start taking her out three-four times a day (bag of treats in one hand) and then, when you know she has done her business, skip the crate (I guess there might be reasons to crate a dog, but as I've said before on this forum, I live in a country where crating your dog is against the law, so to me, crating is very odd and just not an option!). I would restrict the dog to one room (the kitchen, perhaps) and close the doors, until you know she can be trusted and that she feels ok with the house.
And regarding rain - Castor hates rain too, but we still take him out and I refuse to go back in before he has peed. He would never go out into a garden willingly in rain; if it's raining in the morning, he actually refuses to get out of bed so some mornings are a bit of a struggle here too (and this bulldog sleeps on the third floor so lots of stairs he needs to go down... but he knows there is a treat waiting for him when he's got his harness on!).
I know it's difficult to train a bulldog, but treats and kisses can do wonders! And perhaps you have some friend with a nice dog who can meet up with you so the dogs can play or walk together? Castor will hop down the stairs when we tell him that his friends might be in the park.