The monoclinic crystal structure of the undoped BaBiO₃ can be described as a cubic perovskite which is distorted by the frozen breathing and tilting phonon modes of the BiO₆ octahedra. The phonon mode softening is experimentally observed (Braden et al 1996 Europhys. Lett. 34 531) in Ba₁-xKxBiO₃ through potassium doping followed by a transition into an ideal cubic perovskite structure at x = 0.37, close to the emergence of superconductivity. In our previous paper (Korotin et al 2012 J. Phys.: Condens. Matter 24 415603) we demonstrated that it is necessary to take into account correlation effects using the DFT+U method in Wannier functions as a basis to obtain a good agreement between the calculated and experimental values of crystal structure distortion and the energy gap in BaBiO₃. In the present work, using the same method, we calculated the breathing mode phonon frequencies as a function of the potassium doping level in Ba₁-xKxBiO₃. The obtained frequencies are in good agreement with experimental values and the breathing mode softening with doping is reproduced, contrary to calculations made without consideration of correlation effects. We show that the cubic crystal structure becomes stable at x = 0.30 in agreement with the experimental transition to cubic perovskite at x = 0.37. The possible connections between the correlation effects, phonon mode softening and superconductivity in Ba₁-xKxBiO₃ are discussed.