weight constrained shortest path problem variants
Check the definition of reduction again.  A reduction that works for general dags is a perfectly valid reduction.  It meets all of the conditions in the definition.  I suggest working through the definition -- that should help you see what is going on.

Or, to put it another way: The problem for "general DAG" is at least as hard as the problem for "a particular type of DAG", so if you've proved the problem is NP-hard for a particular type of DAG, you've also proven it is NP-hard for general DAGs, too.