OWS not using freestyle.

I gave been struggling with a rotator cuff issue-making freestyle almost impossible. Backstroke, breaststroke, sidestroke, over arm sidestroke (with good arm) are fine. I can make headway, but it does feel like cleaning a gym floor with a toothbrush. Is this practical for ~2-4 mile open water swims? I have an event coming up; and I have 2 (maybe more?) choices. Skip it (kind of important-don't really want to do that) or do it with theses non traditional strokes. I am not too concerned about time (other than falling too far behind and getting pulled from the race). Besides, if I can get through this under these circumstances, I will give myself all kinds of extra credit.