 Solution

For $0\le x\le B$, the two variational comparisons agree. Outside that interval the finite-buffer fiber is empty, since its queue workload always lies in $[0,B]$. The finite-buffer workload rate truncation is thereforeEquivalently this is J for $x\le B$ and infinity above B, since J already assigns infinity to negative queue workloads. Together with the contracted LDP from the first part, this proves the requested principle. The rate is good by contraction; alternatively its finite sublevel sets are the compact sets $\{J\le M\}\cap [0,B]$.