A few issues to address here so I'm going to try to tackle this in parts. Though before that I just want to say thank you for being willing to adopt and care for your daughter.
You seem to presume that birth control (BC from now on) is causing your daughter to have severe menstrual cramping, and I find that unlikely. Hormonal BC is actually prescribed to help relieve cramping and other symptoms of severe period so if anything I would think BC would be helping your daughter's pain.
So first step is to find out what is causing your daughter pain. If it's really that bad she must be taken to a doctor to determine the actual cause of her discomfort. During such a visit you can mention your belief that the cause is birth control and see what the doctor says about it. I suspect the doctor will say the same thing that I did and that BC is more likely to be helping that harming her periods. Still hearing that directly from a doctor might help to relieve your concerns better then anything I can say.
If your doctor does believe the BC is the cause of the problem, which I stress is doubtful, something should be done about it. Though unless your daughter explicitly says she doesn't want to take BC (more on that below) the solution would likely be not to stop BC but to change to one that didn't have undesirable side effects. It's not uncommon to go through a few BC pills until one that works best for the person taking it is found.
Regardless of the cause if your daughter's pain is really that severe you should be seeing a doctor to figure out the potential causes and how it can be fixed. This should be your first step!
What is your daughter's opinion?
You have made clear what you and your wife's view on birth control is, but not what your daughter's was. You said that she was 'coerced', but you didn't clarify what that means. Have you sat down with her and asked her what her thoughts were on taking birth control? Did she express a desire to not take it or a concern that her mother felt so strongly about it?
Even if your wife suggested birth control that alone is not the same thing as 'forcing' it. For it to be forced your daughter needs to actually be upset about taking it. Given how easy BC is to take, the usual minimal amount of side effects, and the potential positives in regulating and minimizing the pain of a period generally means BC can be useful and desirable even if one is not expecting to engage in sexual activity. To be frank it seems unlikely your daughter would be strongly enough opposed to BC for you to need to worry about her feeling coerced.
That being said the first thing to do is to make sure you understand her actual feelings, without allowing either parent to project their views on the child. That means asking her in a neutral manner how she feels about the birth control. Don't ask in a way that suggests an answer you want, let her give the answer she feels. Make sure this is actually a problem in your daughters opinion before you make it one for you and your wife.
Of course if your daughter does express concern about taking BC then ensuring her wishes are respected is important. That would start with a conversation between the three of you and if you really can't agree may require considering a conversation with a doctor or a therapist...more on that as well below...
Is this 'medication' even birth control?
As mentioned in the comments you don't drink birth control. It's also unlikely that mother and daughter would be taking the same BC at the same time of day, and since BC is usually taken first thing in the morning it's doubtful you would witness it.
So what did you see them drinking? If you think what you saw them drinking caused your daughter's cramping, well it would help if we knew what that medication was since it's unlikely to be traditional hormonal birth control. Again if you take your daughter to a doctor about her cramping you can ask about whatever she drunk was and if a doctor would recommend her taking or stopping that medication.
Is your concern really your daughter's autonomy, or birth control itself?
Forgive me for asking but, well you sound a bit hostile to the concept of taking birth control in your question. In fact I feel the need to point out that availability of birth control has shockingly little affect on a teens choice to be 'sexually active' so your concern about your daughter being promiscuous due to BC is a non issue, especially since teens don't usually choose to have sex until 16-17 not 13. I personally would also point out that most experts in teen psychology agree that informed, safe, consensual sex with similarly aged peers is not necessarily harmful if your daughter did choose to engage in sex as she grew older.
I ask about this because how I answer is kind of dependent on what your concern really is. My answer here is sticking with the presumption that your problem is about your daughter's ability to pick medication she takes and not specifically about the medication in question being birth control. If your problem is specifically that it's birth control that would be an entire different question and answer that would be needed.
You and your wife need to talk about your relationship regardless
Separate from your specific question about BC it seems you need talk with your wife about your relationship. You accuse your wife of not listening to you, of using you only for money, of being emotionally manipulative etc. This does not sound like you are happy with your current relationship, and that is a problem in and of itself that affects not just you and your wife, but your daughter as well.
Now being aromantic I'm a terrible person to give relationship advice - you might as well ask a blind person if these jeans make your butt look fat for all the good it would do you. However the one thing I can say is that relationship counseling or speaking to a therapist can help if one or both partners are unhappy in a relationship and it may be worth considering.
In addition to helping with your other relationship problems a therapist could help you speak about your concern with BC and rather your daughter should take it and help you both to come to a better agreement if you still find yourself concerned after speaking with doctor and daughter about it.
I'm afraid that's as much as I can really say without knowing far more about your and your situation. I hope something in all this was useful for you.