It is correct that dispersion forces are not described correctly in DFT. However, DFT can in some cases give rise to semi long range interactions which by coincidence agree with the dispersion forces, see forinstance :
K. Stokbro, E. Nielsen, E. Hult, Y. Andersson, and B. I. Lundqvist, Nature of bonding forces between two hydrogen passivated silicon wafers, Phys. Rev. B. 58, 16118 (1998).
For the case of carbon materials you may check how well gga and lda describes the experimental interlayer distance in graphene. According to my memory lda does the best job, and you can expect the same level of accuracy when using lda to simulate your carbon system.