Multi-material lattice structures offer unprecedented opportunities for designing multifunctional engineering systems, such as lightweight load-bearing components with enhanced thermal management capabilities. However, fully exploiting their design potential requires advanced optimisation frameworks that can balance extensive design freedom with computational efficiency. This work presents a data-driven multiscale optimisation framework for the design of multi-material lattice structures under competing mechanical and thermal objectives. At the macroscale, topology optimisation is employed to determine the spatial distribution of material and microstructural features. Curved body-centered cubic (BCC) unit cells are adopted at the microscale, where strut geometry is defined by cubic spline and local volume fraction. A supervised machine learning (ML) surrogate model is trained to efficiently map these unit cell design parameters to their homogenised properties, accelerating performance evaluation during the iterative optimsiation process. A multi-material interpolation scheme, analogous to SIMP-based approaches, is introduced to control solid-void transitions and material selection. The proposed framework will be benchmarked against existing multi-material topology optimisation (MMTO) methods to demonstrate the performance advantages of multi-material designs, as well as the substantial computational savings achieved through the integration of ML-based homogenisation.