M1 transitions from the $^6$Li($0^+;T=1$) state at 3.563 MeV to the $^6$Li($1^+$) ground state and to the $alpha+d$ continuum are studied in a three-body model. The bound states are described as an $alpha+n+p$ system in hyperspherical coordinates on a Lagrange mesh. The ground-state magnetic moment and the gamma width of the $^6$Li(0$^+$) resonance are well reproduced. The halo-like structure of the $^6$Li$(0^+)$ resonance is confirmed and is probed by the M1 transition probability to the $alpha+d$ continuum. The spectrum is sensitive to the description of the $alpha+d$ phase shifts. The corresponding gamma width is around 1.0 meV, with optimal potentials. Charge symmetry is analyzed through a comparison with the $beta$-delayed deuteron spectrum of $^6$He. In $^6$He, a nearly perfect cancellation effect between short-range and halo contributions was found. A similar analysis for the $^6$Li($0^+;T=1$) $gamma$ decay is performed; it shows that charge-symmetry breaking at large distances, due to the different binding energies and to different charges, reduces this effect. The present branching ratio $Gamma_{gamma}(0^+to alpha+d)/Gamma_{gamma}(0^+to1^+)approx 1.3times 10^{-4}$ should be observable with current experimental facilities.
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