  We propose a method to compute the scattering angle for classical black hole
scattering directly from two massive particle irreducible diagrams in a
heavy-mass effective field theory approach to general relativity, without the
need of subtracting iteration terms. The amplitudes in this effective theory
are constructed using a recently proposed novel colour-kinematic/double copy
for tree-level two-scalar, multi-graviton amplitudes, where the BCJ numerators
are gauge invariant and local with respect to the massless gravitons. These
tree amplitudes, together with graviton tree amplitudes, enter the construction
of the required $D$-dimensional loop integrands and allow for a direct
extraction of contributions relevant for classical physics. In particular the
soft/heavy-mass expansions of full integrands is circumvented, and all
iterating contributions can be dropped from the get go. We use this method to
compute the scattering angle up to third post-Minkowskian order in four
dimensions, including radiation reaction contributions, also providing the
expression of the corresponding integrand in $D$ dimensions.
