Today I had rented a Piper Archer III for a local flight. After taxing to the fuel and filling up, I went to start the plane again and it did not want to turn over. This particular plane is somewhat fussy and sometimes doesn't start right away. As I continued to crank the engine over, you could tell the cranking lulled and the battery was having difficulty turning the engine over as effortlessly as it typically would. At one point the prop wouldn't even rotate.
I eventually got the plane started after a series of cranking / resting periods. I made the decision to not fly given the battery issue and returned the plane to its hangar.
The owner of the flight school was initially reluctant to comp the 1/3 of an hour I had on the hobbs (from taxing to gas pumps, etc) stating that once I had the plane started, I could have flown since it would have been powered by the magnetos.
My question: I know that the plane is powered by the magnetos once started, however, was my decision to not fly unfounded and unnecessarily cautious or would there be good reason not to fly knowing that the battery was weak and unreliable?
Update: It seems that some answers and comments have a focus on the financial aspect of my question. To be clear, this is not about money or how much. No one should pay for goods or services that were not rendered. The FBO's argument was that the choice not to fly was my own and not due to a responsibility of the FBO's to fix a "broken" plane. The money aspect is just an indicator of where the responsibility lies, on my shoulders or the FBO's.