Elastic fingering supplements the already interesting features of the traditional viscous fingering phenomena in Hele-Shaw cells with the consideration that the two-fluid separating boundary behaves like an elastic membrane. Sophisticated numerical simulations have shown that under maximum viscosity contrast the resulting patterned shapes can exhibit either finger tip-splitting or side-branching events. In this work, we employ a perturbative mode-coupling scheme to get important insights into the onset of these pattern formation processes. This is done at lowest nonlinear order and by considering the interplay of just three specific Fourier modes: a fundamental mode n and its harmonics 2n and 3n. Our approach further allows the construction of a morphology diagram for the system in a wide range of the parameter space without requiring expensive numerical simulations. The emerging interfacial patterns are conveniently described in terms of only two dimensionless controlling quantities: the rigidity fraction C and a parameter Γ that measures the relative strength between elastic and viscous effects. Visualization of the rigidity field for the various pattern-forming structures supports the idea of an elastic weakening mechanism that facilitates finger growth in regions of reduced interfacial bending rigidity.