I thought about this thread during my 3-setter this morning. My opponent definitely served at and played to my backhand a higher percentage of the time.
Now, a little different because we were both righties (and fairly equal skill). But, when we got into cross court exchanges I would work hard to step around and hit an inside out FH with more pace than I can produce on my OHBH. My goal was to produce a defensive shot that I could become offensive with and attack with a cross court FH approach. He would be on the run for his FH which made his pass attempts and lobs difficult.
I also had success serving wide and giving myself more court to work with. Some serve and volley, but mostly getting shorter returns that put me on offense. Ultimately, had to be mentally strong on the short balls I got because there weren't many of them.