By measuring the nuclear magnetic resonance (NMR) T_{1}^{-1} relaxation rate in the Br (bond) doped DTN compound, Ni(Cl_{1-x}Br_{x})_{2}-4SC(NH_{2})_{2}(DTNX), we show that the low-energy spin dynamics of its high magnetic field "Bose-glass" regime is dominated by a strong peak of spin fluctuations found at the nearly doping-independent position H^{*}≅13.6  T. From its temperature and field dependence, we conclude that this corresponds to a level crossing of the energy levels related to the doping-induced impurity states. Observation of the local NMR signal from the spin adjacent to the doped Br allowed us to fully characterize this impurity state. We have thus quantified a microscopic theoretical model that paves the way to better understanding of the Bose-glass physics in DTNX, as revealed in the related theoretical study [M. Dupont, S. Capponi, and N. Laflorencie, Phys. Rev. Lett. 118, 067204 (2017).PRLTAO0031-900710.1103/PhysRevLett.118.067204].