I have not seen any one convert regular chair, but I have plans to convert my Quickie GPV if I do not manage to get a real sports chair. Yes sports chairs are very expensive, I participate in games with Clubs sportschair, my own Sports chair will enable me to practice more and adjust chair to my liking

My first tournament 2month after I started playing
Photo1 Photo2Wheelie bar can give enough stability to play badminton, though an active Anti tip is better. In a casual game you can play a lot without stretching far back or front. Make sure you test leaning forward with casters pointed backward as some wheelchairs tip forward when leaning forward. After few games you will learn what shots are safe to play, and what movements you can make safely.
Once a week I practice with my friends on regular wheelchair (none of us have wheelie bar) and we play shots mainly in front, and side. When the shuttle is behind we put body weight forward and hit shuttle by stretching only arm backward. With wheelie bar you will be able to play over head shots.
I have not seen the Invacare A4 chair, but searching on net I found following
Document that indicate that you may have option to increase camber.