We consider the pp → ppρ 0 π 0 and pp → pnρ 0 π + processes at LHC energies. Our description is based on the nonperturbative framework of tensor pomeron and tensor reggeon exchanges. We discuss the DrellHiida-Deck type mechanism with centrally produced ρ 0 meson associated with a very forward/backward πN system. The considered processes constitute an inelastic (non-exclusive) background to the pp → ppρ 0 reaction in the case when only the centrally produced ρ 0 meson decaying into π + π − is measured, the final state protons are not observed, and only rapidity-gap conditions are checked experimentally. We compare our results for the γπ + → ρ 0 π + reaction with the experimental data obtained by the H1 collaboration at HERA. We present several differential distributions for the pp → pnρ 0 π + reaction and estimate the size of the proton dissociative background to the exclusive pp → ppρ 0 process. The ratio of integrated cross sections for the inelastic pp → pN ρ 0 π processes, where pN ρ 0 π stands for pnρ 0 π + plus ppρ 0 π 0 , to the reference reaction pp → ppρ 0 is of order of (7-10)%. We present also the ratios of the ρ 0 rapidity and transverse momentum distributions for the inelastic pp → pN ρ 0 π versus the elastic pp → ppρ 0 reaction. Our results may be used to investigate the γπ → ρ 0 π process at LHC energies.