  We consider a system of particles performing a one-dimensional dyadic
branching Brownian motion with space-dependent branching rate, negative drift
$-\mu$ and killed upon reaching $0$. More precisely, the particles branch at
rate $r(x)=(1+f(x))/2,$ where $f$ is a compactly supported and non-negative
smooth function and the drift $\mu$ is chosen in such a way that the system is
critical in some sense. This particle system can be seen as an analytically
tractable model for fluctuating fronts, describing the internal mechanisms
driving the invasion of a habitat by a cooperating population. Recent studies
from Birzu, Hallatschek and Korolev suggest the existence of three classes of
fluctuating fronts: pulled, semi pushed and fully pushed fronts. Here, we focus
on the fully pushed regime. We establish a Yaglom law for this branching
process and prove that the genealogy of the particles converges to a Brownian
Coalescent Point Process using a method of moments. In practice, the genealogy
of the BBM is seen as a random marked metric measure space and we use spinal
decomposition to prove its convergence in the Gromov-weak topology. We also
carry the spectral decomposition of a differential operator related to the BBM
to determine the invariant measure of the spine as well as its mixing time.
