The problem here is the term birth control. Your friend is correct that you may not use birth control for any reason whatsoever. That is not however what your doctor is considering having you do. Your doctor is thinking about prescribing for you medication commonly used for birth control purposes, but is being used in your case to treat painful menstrual symptoms. Taking that medication for legitimate non-contraceptive purposes is not sinful, even if that medication has what would be in your case the unintended side effect of rendering you temporarily infertile. Unfortunately, doctors and the public alike have become used to referring to this particular type of medication as birth control, even when the medication is being used for non-contraceptive purposes.
That said, it may be helpful to seek out a second opinion. It may be that there are other treatments available to treat your menstrual symptoms that would not require you to use this particular medication, and would thereby spare you the dangerous side effects to which the medication can expose you (read the package inserts for the common "birth control" drugs for more details). For more information on alternative treatments, I recommend either contacting or asking your doctor to consult with the Pope Paul VI Institute and/or Culture of Life Family Services.