Lucas recurrences and Lambert series in coprime double sums

For the Lucas numbers $L_0=2$, $L_1=1$, and $L_n=L_{n-1}+L_{n-2}$, we obtain \[ \sum_{\substack{n,m\ge1\\(n,m)=1}}\frac{1}{L_{2n}L_{2m}L_{2(n+m)}}=\frac1{45}. \] This follows from the absolutely convergent identity \[ \sum_{\substack{n,m\ge1\\(n,m)=1}}\frac{q^{n+m}} {(1+q^n)(1+q^m)(1+q^{n+m})} =\fra