Suppose that we take the Calculus of Constructions as a basis, but take away exponential functions (allowing only linear functions), and add the controlled duplication rules of EAL. That'd, I believe, give us a simple core with dependent types that can be reduced on the oracle-free fragment of Lamping's abstract algorithm.
Now, it has been shown  that adding type-level fixed points to EAL do not affect its complexity bounds. Doing so would make the language more interesting, as one would be able to define induction and inductive types. Problem is, the Calculus of Constructions makes no distinctions between types and values. As such, adding Fix would imply its existence on the value level (except if we make an exception of the rule, which looks inelegant), which may cause non-termination.
As such, it is not obvious to me how to add Fix to the language of the first paragraph without inhabiting all types. Perhaps a restricted version of it, with fix variables only occurring on non-application positions (i.e., only as arguments) would suffice, but I'm not sure 1. if that's true, 2. if that'd be the right way.
 J.-Y. Girard. Light linear logic. Information and Computation, 143:175–204, 1998.