Here is a sketch of what is known in essentially arbitrary monoidal‐category–theoretic generality. Nothing really “mystical’’ happens once you drop semisimplicity – you simply rediscover the usual bicategorical/Morita‐theory machinery, now incarnated inside Cat (or inside whatever ambient 2-category you like).

1. Modules and bimodules as 1-cells in a bicategory of monoids
   – A monoidal category C is nothing but a pseudomonoid in the bicategory Cat. A (left) C–module category is then just a pseudomodule over that pseudomonoid; a C–C-bimodule is a 1-cell in the bicategory of pseudomonoids.
   – All of this is spelled out in the “formal theory of monads’’ / “doctrines’’ literature:
   • Bénabou, “Distributors at Work’’ (the original bicategory–of–profunctors setup).
   • G. M. Kelly, “Basic Concepts of Enriched Category Theory’’ (especially the parts on coends and on the Eilenberg–Watts theorem in the enriched setting).
   • R. Street, “Monoidal bicategories and Hopf algebroids’’ (in Quantum Groups, LNM 1510)– this shows how to build the bicategory C–Bimod, and exhibits the relative tensor product as a bi(co)end.

2. The relative tensor product over C
   – If C admits those reflexive coequalizers (or more generally coinserters/coequalizers) that the tensor product ⊗C must preserve in each variable, then for any pair M∈C–Mod, N∈Mod–C one can form the coequalizer
   M⊗C⊗N ⇒ M⊗N → M⊠_C N
   in Cat. This coequalizer is the “tensor product of module categories over C,’’ and makes C–Bimod into a monoidal bicategory with unit C itself.
   – References for existence and basic properties of that coequalizer construction can be found in
   • G. M. Kelly, “On the Eilenberg–Watts theorem in enriched category theory’’ (J. Algebra 1982)
   • R. Street, “Frobenius monads and pseudomonoids’’ (in Fields Institute volume on Hopf algebras)
   • M. Shulman, “Framed bicategories and monoidal fibrations’’ (for a modern treatment of coends in bicategories).

3. Morita theory / Eilenberg–Watts in general
   – Just as in ordinary algebra, any cocomplete C–module category M which admits a “progenerator’’ X can be shown (by the usual density argument / monadicity) to be equivalent as a C–module to A–Mod₍C₎ for the monoid A=End_C(X).
   – This is the enriched/Eilenberg–Watts / formal‐monad‐theory result, proved for instance in
   • R. Street, “The formal theory of monads” (J. Pure Appl. Algebra 1972)
   • G. M. Kelly and R. Street, “Review of the elements of 2-categories’’ (Lecture Notes in Mathematics, 1974)
   • P. Lack, “A coherent approach to pseudomonads’’ and follow-up papers of Lack–Street on bicategorical monad theory.

4. Azumaya algebras and invertible module categories
   – In a braided monoidal category C one defines a monoid A to be Azumaya if A⊗– establishes an equivalence C→A–Bimod₍C₎. One then shows exactly as in the classical ring‐theoretic case that A–Mod is an invertible C–module category, with inverse given by Mod–A.
   – The group of equivalence‐classes of such Azumaya algebras is the “internal Brauer group’’ Br(C). Moreover, when you extract from the monoidal bicategory C–Bimod only the invertible 1-cells (bimodule‐categories), 2-cells (bimodule‐functors) and 3-cells (natural transformations) you recover a 3-group (in fact a bigroupoid) which one can identify with the Brauer–Picard 3-group of C.
   – This circle of ideas appears in
   • B. Pareigis, “Brauer groups in monoidal categories’’ (Manuscripta Math. 1979)
   • H.-J. Schneider and P. Schauenburg, “Hopf algebroids and quantum groupoids’’
   • A. Bruguières & A. Virelizier, “Hopf monads on monoidal categories’’ (Adv. Math. 2007)

5. The finite (nonsemisimple) tensor‐category case
   – All of the above goes through when C is a finite tensor category (finite abelian, rigid, k-linear). In that context one adds the hypothesis that modules are exact functors, etc., but no semisimplicity is needed.
   – In particular every exact indecomposable C–module has a progenerator, hence comes from some algebra in C, and invertible modules are exactly those coming from Azumaya algebras.
   – The cleanest presentation of the Brauer–Picard 3-group in this setting is in
   • P. Etingof, D. Nikshych & V. Ostrik, “Fusion categories and homotopy theory’’ (Quantum Topol. 2010)
   • The survey “Finite tensor categories’’ by Etingof–Gelaki–Nikshych–Ostrik (2005)

6. Summary of the up-shot
   – In complete generality you have a monoidal bicategory C–Bimod whose 1-objects are C–C-bimodule categories, whose 1-cells are bimodule functors, etc. Composition is “tensor over C.’’
   – A Morita theorem (via monadicity) says that every suitably cocomplete C–module with a generator is equivalent to modules over some algebra in C.
   – In the braided case, Azumaya algebras are precisely those algebras whose module categories are invertible in that bicategory, and one recovers exactly the Brauer–Picard 3-group.

So nothing “mystical’’ is hiding once you leave semisimplicity – you simply run the well-known bicategorical/Morita‐theory machinery in the ambient 2-category Cat (or in k-linear Cat, or in whatever enrichment you prefer). The key references above will show you (a) how to form the relative tensor product, (b) how to prove the general Eilenberg–Watts result, and (c) how to identify invertible modules with Azumaya algebras and hence get your Brauer–Picard 3-group.