The key is showing your address, like you said you're sending info about your bank account, make sure you get a statement with the address on it. Have you done any online shopping? Any receipts with shipping information to the address with your name? Any mail addressed to you at the address would be something to prove the address, even if it holds very little weight. Uber or food delivery receipts on your phone or email showing the address? We included mail from a gym membership and car insurance I had. Does your partner have a car? Can he make you a joint owner? What about his car insurance? I've heard of others who sent in grocery store reward members mail. Get your friends to send you some mail at your address. Are all your utilities included in rent? If not, did you try contacting all the companies to add a secondary person? What about internet? Did you try to get a driver's licence? What province are you in? I'm the sponsor and I changed by BC Medical Services Plan to say I'm in a common-law relationship. Go and get wills done? See if you can get a local library card or leisure centre card with your name on it? It's about showing you have made a home in the area, so it's good you're going to send in the hospital records so it shows you've been in the area. You said before you had a lease statement that said 4 people were living in the house, even if it doesn't provide names it's again something else to include. Why don't you get a Statue of Declaration that you're in a common law relationship?
https://www.canada.ca/content/dam/ircc/migration/ircc/english/pdf/kits/forms/imm5409e.pdf
Don't give up hope yet! I know some of this sounds farfetched but if you think you're going to be refused anyway, don't you think its worth a shot to try it?