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We study new classes of generic off-diagonal and diagonal cosmologicalsolutions for effective Einstein equations in modified gravity theories, MGTs,with modified dispersion relations, MDRs, encoding possible violations of(local) Lorentz invariance, LIVs. Such MGTs are constructed for Lagrangedensities with two non-Riemannian volume forms (two-measure theories, TMTs) andassociated bimetric/ biconnection geometric structures. For conventional 2+2splitting, we can describe such models in Finsler like variables, which isimportant for elaborating geometric methods of constructing exact andparametric solutions. Such formulations of general relativity, GR, and MGTs areconsidered for Lorentz manifolds and their (co) tangent bundles, in brief,FTMT. Off-diagonal metrics solving gravitational field equations in FTMTs aredetermined by generating functions, effective sources and integration constantsand characterized by nonholonomic frame torsion effects. Restricting the classof integration functions, we can extract torsionless and diagonalconfigurations and model emergent cosmological theories with square scalarcurvature, $R^2$, when the global Weyl-scale symmetry is broken via nonlineardynamical interactions with nonholonomic constraints. In the physicalEinstein-Finsler frame, the constructions involve (i) nonlinearre-parametrization symmetries of the generating functions and effectivesources; (ii) effective potentials for the scalar field with possible two flatregions which allows a unified description of locally anisotropic and/orisotropic early universe inflation related to acceleration cosmology and darkenergy; (iii) there are "emergent universes" described by (off-) diagonalsolutions for certain nonholonomic phases and parametric cosmological evolutionresulting in various inflationary phases; (iv) we can reproduce in two-measuretheories massive gravity effects.

Broken Scale Invariance, Gravity Mass, and Dark Energy in Modified Einstein Gravity with Two Measure Finsler Like Variables