You can use AsymptoticSolve for this purpose:

AsymptoticSolve[h == H[L, c], {L, 1/Sqrt[-2 h]}, {c, Infinity, 5}]

{{L -> 1/(Sqrt[2] Sqrt[-h]) + (12 Sqrt[2] Sqrt[-h] + 15 h J - h J m)/( 4 Sqrt[2] c^2 Sqrt[-h] J) - (-280 Sqrt[2] - 240 Sqrt[2] h J^2 + 35 Sqrt[-h] h J^3 + 80 Sqrt[2] m + 96 Sqrt[2] h J^2 m + 30 Sqrt[-h] h J^3 m + 3 Sqrt[-h] h J^3 m^2)/( 32 Sqrt[2] c^4 J^3)}}

You can use AsymptoticSolve for this purpose:

Lookup[L] @ First @ AsymptoticSolve[h == H[L, c], {L, 1/Sqrt[-2h]}, {c, Infinity, 5}] +O[c,Infinity]^5 //TeXForm

$ \frac{1}{\sqrt{2} \sqrt{-h}}+\frac{-h J m+15 h J+12 \sqrt{2} \sqrt{-h}}{4 \sqrt{2} c^2 \sqrt{-h} J}-\frac{3 \sqrt{-h} h J^3 m^2+30 \sqrt{-h} h J^3 m+35 \sqrt{-h} h J^3+96 \sqrt{2} h J^2 m-240 \sqrt{2} h J^2+80 \sqrt{2} m-280 \sqrt{2}}{32 c^4 \left(\sqrt{2} J^3\right)}+O\left(\left(\frac{1}{c}\right)^5\right) $