What is the importance of a TCS researcher in a mostly applied CS department? Stating the title in another way, how to convince a strong applied CS department that a theorist would be a better choice than one more applied computer scientist?

I understand that it is natural for a applied department to want one more expert on a specific field. On the other hand, diversity is important and I think the importance of a theorist goes beyond better theory courses.

What is the importance of a TCS researcher in a mostly applied CS department?
I am really not qualified to give an informed answer to your question, but there are many reasons for which a theory-oriented researcher might be a good fit for a more applied environment.

The main reason that comes to mind is this: good theoretic foundations are invaluable for a thorough understanding of an application which in turn increases trust, efficiency and decreases mental effort. In general it allows work to proceed much faster, as the foundations provide a mental reference that can be easily extended or made explicit.

I'm sure there are many examples in complexity theory, where an abstract approach has led to fantastic practical improvements in concrete implementations, but the field I am the most familiar with is "theory B", in which programing language design has greatly benefited from advances in pure computing theory research.

Haskell is a particularly good example. On the theory side, we have simple core languages like system $\mathrm{F}_\omega$ which have proven to be a very useful base for the core language of the GHC compiler. The research in type inference/reconstruction and more recently, on dependent types has guided the foundations of the Haskell design, including the Hindley-Milner architecture and GADTs. Monads, of course, were introduced in the from the most abstract domain of mathematics to the computing world by Moggi, a theorist.

More pragmatically, knowledge of the theory of operational semantics turns out to be an invaluable tool for language extensions: the paper which introduces Software Transactional Memory in Haskell has explicit operational semantics for the small sub-language containing the STM constructs. This creates a reference for the implementation, and the ability to prove properties about a correct implementation, creating trust and understanding.

All this point to the fact that having a strong theory background is an important asset of every research team, including those with a strong application theme.