In earlier work by den Hollander, König, and dos Santos, the asymptotics of the total mass of the solution to the parabolic Anderson model was studied on an almost surely infinite Galton-Watson tree with an i.i.d. potential having a double-exponential distribution. The second-order contribution to this asymptotics was identified in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The present paper extends this work to the degree-normalised Laplacian. The normalisation causes the Laplacian to be non-symmetric and which leads to different spectral properties. We find that the leading order asymptotics of the total mass remains the same, while the second-order correction coming from the variational formula is different. We also find that the optimiser of the variational formula is again an infinite tree with minimal degrees. Both of these results are shown to hold under much milder conditions than for the regular Laplacian.