It looks like you're already doing a 2-beat kick.
By the way, I went to the pool today and tried something different. Instead of making two big kicks (to simulate the glide phase) between switching from one side to the other, I decided to glide by eliminating the kicks or kick very little. To my surprise, I was able to reduce the stroke count to 16 without any problem! I guess the two big kicks I was doing before were creating so much drag that it increased the stroke count. Now my backstroke stroke count exactly matches my freestyle stroke count, which makes me a happy camper. :-) I don't know if gliding this way is considered cheating, but I'm satisfied for now.
Thanks everyone for your help