Shortest path in a DAG consisting of multiple copies of a smaller DAG
Yes, you certainly can (based on the fact that any subpath of a minimal path must also be minimal). That is, any shortest path entering $H_i$ at $u$ and leaving at $v$ must follow the shortest path from $u$ to $v$ in $H_i$.

Basically, you can compute the shortest-path distance matrix $D$ for any $H_i$ (it would be the same for all of them, of course), and replace every subgraph $H_i$ by one consisting only of the in- and out-nodes (that is, the nodes connected to other subgraphs; presumably fewer than $|V_i|$), and use only direct edges from the in-nodes to the out-nodes, with weights given by $D$.

You don't need to explicitly construct this new graph, of course. If you have the macrostructure of $G$ available in implicit form, you can compute $D$, and use that together with the macrostructure of $G$ in a (slightly customized) DP algorithm for finding the shortest path.