According to my research they do not react with one another and something like Anavar, primo, or winstrol will not counteract the birth control pills.
The bigger issue is you will not see the results you are looking for as the BC is raising estrogen which unfortunately for ladies can cause weight gain and water retention. Also driving up estrogen will counter the higher testosterone you are trying to put into your body to make muscle gains. I'm not saying that you won't build any muscle, but you also run the risk of gaining some fat in the process.
To get better results other BC options should be considered. If you and your partner have no desire to have children in the future the options open up, vasectomy and ablation surgeries are both quick with relatively short recovery times. If you are waiting to have children later in life you are very limited. On the upside for women, two different birth control pills for men have passed the first clinical trials and are moving onto the second set of clinical trials which involve increasing the number of participants. It would be a long shot, but your partner may be able to write the pharmacies and get in on the second human trial if they feel like being a Guinee Pig.