I've had a really bad relationship with my supervisor since I started. I've even had to take time off to get a mental reprieve from everything that was happening. At this point, I really want to go somewhere else. This is not an American university, and I'm in the social sciences. In order to apply for admissions, you have to have the support of your prospective supervisor. I know the person that I would like to have as my supervisor at the other university and they sponsored me as a visiting student at their uni for a semester. However, this person has a strong relationship with my current supervisor. My concern would be that the prospective supervisor wouldn't want to jeopardize their relationship with their colleague by taking me on and that reaching out to them will jeopardize my ability to get a recommendation from them after I graduate. I'm also concerned that this would further strain my already intolerable relationship with my current supervisor, as I'm guessing that the prospective supervisor would reach out to the current one. For background, my research is progressing, and I don't have research or academic-related problems. It's really that my supervisor and my department are unsupportive to downright abusive.
Your concern is correct: If your prospective supervisor supported your application and adopted you as a PhD student, that might put a significant strain on their relationship to your current supervisor. It seems unlikely that your prospective supervisor would want to risk that. You can talk to them about your idea, but be prepared for a negative response.
Your best bet would be to look for another supervisor who doesn't have any noteworthy relationship to your current one.
This is unfortunate, and I would advise you not to collaborate with the potential PhD supervisor. At best, it will lead to awkwardness, at worst to your current situation to deteriorate rapidly. You should find someone neutral.
However, you should maintain careful contact with the potential supervisor, as his/her behaviour towards you sounds decent. The contact can be useful and after you move your former supervisor does no longer affect you. Some information may seep towards him, and I fully understand the difficulty of being honest without disclosing the real situation, but it can also be beneficial in the mid-term.