Is this a good essay premise for Bryn Mawr?

As I read your question, you decided that the act of applying (that is, filling out the application) for a women’s college would be awful. I do not think that is a good message to convey as the college probably does not care whether you think that the application process was boorish. No one likes the college application process. You need to convey as honestly as possible how you think that you would benefit from that college and how the college might benefit from having you as a student there. The college application for a women’s college is probably not too different from an application to a coeducational school aside from checking the box to affirm that you are female and thus meet the key demographic requirement for attending a women’s college.