Tsunami observations have particular importance for resolving shallow offshore slip in finite-fault rupture model inversions for large subduction zone earthquakes. However, validations of amplitude linearity and choice of subfault discretization of tsunami Green's functions are essential when inverting tsunami waveforms. We explore such validations using four tsunami recordings of the 25 October 2010 Mentawai Mw 7.8 tsunami earthquake, jointly inverted with teleseismic body waves and 1 Hz GPS (high-rate GPS) observations. The tsunami observations include near-field and far-field deep water recordings, as well as coastal and island tide gauge recordings. A nonlinear, dispersive modeling code, NEOWAVE, is used to construct tsunami Green's functions from seafloor excitation for the linear inversions, along with performing full-scale calculations of the tsunami for the inverted models. We explore linearity and finiteness effects with respect to slip magnitude, variable rake determination, and subfault dimensions. The linearity assumption is generally robust for the deep water recordings, and wave dispersion from seafloor excitation is important for accurate description of near-field Green's functions. Breakdown of linearity produces substantial misfits for short-wavelength signals in tide gauge recordings with large wave heights. Including the tsunami observations in joint inversions provides improved resolution of near-trench slip compared with inversions of only seismic and geodetic data. Two rupture models, with fine-grid (15 km) and coarse-grid (30 km) spacing, are inverted for the Mentawai event. Stronger regularization is required for the fine model representation. Both models indicate a shallow concentration of large slip near the trench with peak slip of ~15 m. Fully nonlinear forward modeling of tsunami waveforms confirms the validity of these two models for matching the tsunami recordings along with the other data.