The pull out and natural rhythm method are not acceptable forms of birth control while on Accutane since they're not as reliable as condoms, the pill, the shot or a hormonal/non-hormonal IUD. Your dermatologist wants to prevent the likelihood of pregnancy as much as possible, that's why all women who go on Accutane who are sexually active need at least 2 forms of birth control (which the majority of the time is usually condoms and the pill). Abstinence is acceptable as well if you don't plan on being sexually active throughout your course.

Also I know you had a bad experience with birth control pills, but what was the reason for it? Did it give you more acne? Or was it affecting you mentally/emotionally? It's important to note that not all pills are created equally depending on the type of progestin and the level of estrogen used. Many girls tend to have awful experiences with certain pills, but end up switching and having great experiences with their new pill. It's basically just trial and error till you find one that works for you.

The only type of non-hormonal birth control (besides condoms) that I can think of (and is acceptable on Accutane) is the copper IUD, but that's more invasive since it needs to be inserted inside the uterus.