How to test balance between right and left channel

SPL meter is probably different from your ears. Since human hearing is almost never balanced perfectly between left and right ear, the surest test is by ear IMHO.

I've spent years trying to discover the source of this problem. Since everything but the room has been eliminated, that must be the problem. Although with left and right volume controls, the problem is easily corrected; like a foolish "audiophile", I still wonder what the source of the problem is.

Well then, let me enlighten you. It's not the room. It can exacerbate the problem, but it is never the source. The OP is on the right track. It absolutely has to do with signal level as it relates to the output devices within your amp. If both channels aren't set 'exactly' the same, this problem is the result. I'm referring to both bias and dc offset. You've probably never heard it functioning properly. And if you've taken your amp in to have it biased, you probably didn't notice any difference since sonic issues are never attributed to these functions by even the tech that spent no more than a half hour on the bench with it. In reality, it takes days or even weeks to properly set these adjustments because the circuit has to settle down and numerous adjustments are necessary until it reaches spec, finally. First bias, then offset, every time. Until you actually do this procedure, you will never realize the potential of your gear, any of it. That's when you will recognize just how little an influence all the other things you've been doing actually has on your system.