The large data sample of the $B_c$ meson collected at the LHC experiment and the HL-LHC experiment provides us the opportunity to study the $B_c$ decays and the related physics. In this paper, we investigate the effect of $K^0-bar{K}^0$ mixing on the the branching ratios, CP violations and CPT violations in the $B_{c}^{pm}rightarrow B^{pm} K_{S,L}^{0}$ decays. We find that some of the $B_c^{pm}rightarrow B^{pm} K_{S,L}^0rightarrow f_{B^{pm}} f_{K_{S,L}^0}$ decay chains have large branching ratios, whose maximum value can exceed the order of $10^{-6}$, the minimum number of $B_c^pm$ events times efficiency for observing the decays at three standard deviations (3$sigma$) level is about $ 10^6$. We study the CP asymmetries in the $B_c^{pm}rightarrow B^{pm} K_{S,L}^0$ decays and find that the CP asymmetries can exceed the order of $10^{-3}$, which are dominated by $K^0-bar{K}^0$ mixing. We give the most promising processes to observe the CP violations and the ranges of the numbers of $B_c^pm$ events-times-efficiency needed to observe the CP asymmetries at a significance of 3$sigma$ in these decays. We investigate the possibility to constraint the CPT violation parameter $Re(z)$ in the $B_c^{pm}rightarrow B^{pm} K_{S,L}^0rightarrow f_{B^{pm}} f_{K_{S,L}^0}$ decays and give the most promising processes to extract the parameter $Re(z)$. We find that the sensitivity for the measurement of parameter $Re(z)$ can reach below $10^{-3}$ in the most promising decays if we assume the selection efficiency is $10^{-3}$ and the total number of $B_c^{pm}$ events collected by the LHCb experiment is $10^{12}$ in the HL-LHC era.
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