I have a non-compromise suggestion first: Have you tried getting off the birth-control pills and using a different method of BC?
Most people on this board who have found ways to compromise have used one of two types. The first is to find a sexual frequency between the two partners' preferences. For example, if one partner wants sex every day (7 times a week) and the other wants it only once a week, then having sex 4 times a week means the HL sacrifices by having sex 3 fewer times a week and the LL sacrifices by having sex 3 more times a week. The compromises aren't always even (e.g., 3 for 3).
Others talk about trading weeks in which each partner gets to determine the frequency of sex. For example, based on the desires of each partner, they may have sex every day for one week and only once the following week.
I, personally, would prefer the latter approach because it might let each get to control not only frequency but also style, intensity, positions, location, props, clothing, etc. Some HL's are not only frustrated about a lack of frequency, but a lack of other qualities they prefer, including variety. The latter type of compromise might resolve that issue as well.