We performed a laser angle-resolved photoemission spectroscopy (ARPES) study on a wide doping range of the ${\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{K}}_{x}{\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ (BaK) iron-based superconductor. We observed a robust low-binding energy (BE) kink structure (kink1) in the dispersion which is doping dependent whereby its energy peaks at the optimally doped level $(x\ensuremath{\sim}0.4)$ and decreases towards the underdoped and overdoped sides. We attribute this kink to electron-mode coupling in good agreement with the inelastic neutron scattering (INS) and scanning tunneling microscopy (STM) results on the same compound where a similar bosonic mode associated with spin excitations was observed. The relation between the mode energy $(\ensuremath{\Omega})$ and the SC transition temperature $({T}_{c})$ deduced from our laser ARPES data follow the universal relation deduced from INS and STM. In addition, we could resolve another kink at higher BE (kink2) showing less doping and temperature dependence compared to kink1 and which thus may be of different origin.