I had the same problem.
It turned out that my daughter had a chronic throat abscess which resulted in her having her tonsils out at 10 months. When she first become fussy on the left side I sought advice from my friend who is a MCHN, a lactation consultant and the ABA helpline but no-one suggested that it may be a medical problem so perhaps make sure that there is not physical issue first.
Eventually the problem was recognised and I've got her back on both sides but for quite a few months she would mainly take the left side. I kept up the supply on the right by pumping, feeding her standing and rocking, getting her while sleepy, lots of skin to skin contact and the football hold.
Conventional wisdom says to offer the least popular side first but this never works for us.