This may be a suggestion from left field, but is sounds like the problem is that the punch holes tear out because all the force from turning the page gets applied at the ring hole which isn't that strong, even with reinforcement, especially when the book is over-filled.

Perhaps what you need to do is look into a different kind of binding technique, and drop the 3-ring approach altogther. I don't know what its called, but there's the method that has the row of rectangular holes in the page and a plastic piece with lots of rings threaded in the whole length of the back (Professors notes printed up by Kinko's often come this way). There's also spiral binding. You could even go so far as to have a book made, or a set of them. I'm sure Kinko's or someone at the local library knows who does binding work.

Of course these are more permanent and less amenable to the adding and subtracting of pages as desired.

On the other hand, you could have a more permanently bound and durable 'Core' book of songs, and then use the 3-ring for collecting new songs. After a while you could recompile an updated core book and have rebound.

Just a thought.