We report the high-pressure properties of two heterometallic perovskite-type metal-organic frameworks (MOFs) templated by dimethylammonium (NH2(CH3)2, DMA+) with the general formula [DMA]MI0.5CrIII0.5(HCOO)3, where MI = Na+ (DMANaCr) and K+ (DMAKCr). The high-pressure Raman scattering studies show crystal instabilities in the 4.0-4.4 GPa and 2.0-2.5 GPa ranges for DMANaCr and DMAKCr, respectively. The mechanism is similar in the two compounds and involves strong deformation of the metal-formate framework, especially pronounced for the subnetwork of CrO6 octahedra, accompanied by substantial compressibility of the DMA+ cations. Comparison with previous high-pressure Raman studies of sodium-chromium heterometallic MOFs show that the stability depends on the templated cation and increases as follows: Ammonium < imidazolium < DMA+. Density functional theory (DFT) calculations are performed to get a better understanding of the structural properties leading to the existence of phase transitions. We calculate the energy of the hydrogen bonds (HBs) between the DMA+ cation and the metal formate cage, revealing a stronger interaction in the DMAKCr compound due to a HB arrangement that primarily involves the energetically preferred bonding to KO6 octahedra. This material however also has a smaller structural tolerance factor (TF) and a higher vibrational entropy than DMANaCr. This indicates a more flexible crystal structure, explaining the lower phase transition pressure, as well as the previously observed phase transition at 190 K, which is absent in the DMANaCr compound. The DFT high-pressure simulations show the largest contraction to be along the trigonal axis, leading to a minimal distortion of the HBs formed between the DMA+ cations and the metal-formate sublattice.