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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 athree-body model. The bound states are described as an $\alpha+n+p$ system inhyperspherical coordinates on a Lagrange mesh. The ground-state magnetic momentand the gamma width of the $^6$Li(0$^+$) resonance are well reproduced. Thehalo-like structure of the $^6$Li$(0^+)$ resonance is confirmed and is probedby the M1 transition probability to the $\alpha+d$ continuum. The spectrum issensitive to the description of the $\alpha+d$ phase shifts. The correspondinggamma width is around 1.0 meV, with optimal potentials. Charge symmetry isanalyzed through a comparison with the $\beta$-delayed deuteron spectrum of$^6$He. In $^6$He, a nearly perfect cancellation effect between short-range andhalo contributions was found. A similar analysis for the $^6$Li($0^+;T=1$)$\gamma$ decay is performed; it shows that charge-symmetry breaking at largedistances, due to the different binding energies and to different charges,reduces this effect. The present branching ratio $\Gamma_{\gamma}(0^+\to\alpha+d)/\Gamma_{\gamma}(0^+\to1^+)\approx 1.3\times 10^{-4}$ should beobservable with current experimental facilities.

Gamma-delayed deuteron emission of the 6Li (0+;T=1) halo state