You've happened upon a very tricky issue indeed! I've got a couple of papers to point you to.
It has been pointed out by D. Judge1 that [itex]L_3[/itex] is not Hermetian on a space of functions that have finite norm on a finite interval. Since [itex]\phi\in(0,2\pi )[/itex], that is exactly what we are dealing with here. Since you tacitly used the Hermiticity of [itex]L_3[/itex] to evaluate those matrix elements, there is a flaw in your argument.
I didn't think of this myself, it is mentioned as a footnote in Problem 9.15 in Liboff's Introductory Quantum Mechanics. In that problem he brings up another inconsistency issue with this commutator (maybe I'll post it if you're sufficiently interested and and I'm feeling sufficiently ambitious). He mentions that W. Louisell1 has pointed out that "more consistent angle variables are [itex]\sin(\phi)[/itex] and [itex]\cos(\phi)[/itex]."
1D. Judge, Nuovo Cimento 31, 332 (1964)
2W. Louisell, Phys. Lett. 7, 60 (1993)
Hope that helps,
You are taking traces of products of infinite dimensional matrices. When you do this, you must be careful about conditional convergence. If you are not using the coordinate represntation, then what do you mean by [itex]\phi[/itex]? You must first express [itex]\phi[/itex] in the [itex]l,m[/itex] basis, and then you will see that your "coordinate independent" conclusion is not so obvious.
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