Improving the interface between copper and carbon nanotubes (CNTs) offers a straightforward strategy for the effective manufacturing and utilisation of Cu-CNT composite material that could be used in various industries including microelectronics, aerospace and transportation. Motivated by a combination of structural and electrical measurements on Cu-M-CNT bimetal systems (M = Ni, Cr) we show, using first principles calculations, that the conductance of this composite can exceed that of a pure Cu-CNT system and that the current density can even reach 1011 A cm-2. The results show that the proper choice of alloying element (M) and type of contact facilitate the fabrication of ultra-conductive Cu-M-CNT systems by creating a favourable interface geometry, increasing the interface electronic density of states and reducing the contact resistance. In particular, a small concentration of Ni between the Cu matrix and the CNT using either an "end contact" and or a "dot contact" can significantly improve the electrical performance of the composite. Furthermore the predicted conductance of Ni-doped Cu-CNT "carpets" exceeds that of an undoped system by ∼200%. Cr is shown to improve CNT integration and composite conductance over a wide temperature range while Al, at low voltages, can enhance the conductance beyond that of Cr.