The decays B+c→J/ψD+sBc+→J/ψDs+ and B+c→J/ψD∗+sBc+→J/ψDs∗+ are studied with the ATLAS detector at the LHC using a dataset corresponding to integrated luminosities of 4.9 and 20.6 fb−1−1 of pp collisions collected at centre-of-mass energies s√=7s=7 TeV and 8 TeV, respectively. Signal candidates are identified through J/ψ→μ+μ−J/ψ→μ+μ− and D(∗)+s→ϕπ+(γ/π0)Ds(∗)+→ϕπ+(γ/π0) decays. With a two-dimensional likelihood fit involving the B+cBc+ reconstructed invariant mass and an angle between the μ+μ+ and D+sDs+ candidate momenta in the muon pair rest frame, the yields of B+c→J/ψD+sBc+→J/ψDs+ and B+c→J/ψD∗+sBc+→J/ψDs∗+, and the transverse polarisation fraction in B+c→J/ψD∗+sBc+→J/ψDs∗+ decay are measured. The transverse polarisation fraction is determined to be Γ±±(B+c→J/ψD∗+s)/Γ(B+c→J/ψD∗+s)=0.38±0.23±0.07Γ±±(Bc+→J/ψDs∗+)/Γ(Bc+→J/ψDs∗+)=0.38±0.23±0.07, and the derived ratio of the branching fractions of the two modes is B+c→J/ψD∗+s/B+c→J/ψD+s=2.8+1.2−0.8±0.3BBc+→J/ψDs∗+/BBc+→J/ψDs+=2.8−0.8+1.2±0.3, where the first error is statistical and the second is systematic. Finally, a sample of B+c→J/ψπ+Bc+→J/ψπ+ decays is used to derive the ratios of branching fractions B+c→J/ψD+s/B+c→J/ψπ+=3.8±1.1±0.4±0.2BBc+→J/ψDs+/BBc+→J/ψπ+=3.8±1.1±0.4±0.2 and B+c→J/ψD∗+s/B+c→J/ψπ+=10.4±3.1±1.5±0.6BBc+→J/ψDs∗+/BBc+→J/ψπ+=10.4±3.1±1.5±0.6, where the third error corresponds to the uncertainty of the branching fraction of D+s→ϕ(K+K−)π+Ds+→ϕ(K+K−)π+ decay. The available theoretical predictions are generally consistent with the measurement.