Setup an EBGP multihop session off your route reflectors, assuming they
are bird, and export all routes and import none on your session facing his
router.
So your customer has an extra bgp session (or 2) for the full route feed.
Your primary bgp session works as is, the multihop sessions
debian stretch, bird v 1.6.3-2 is rock solid
FWIW i saw zero leaks or issues on debian wheezy, jessie, devuan jessie
either.
on multiple rrs taking several million routes I am at:
birdc
BIRD 1.6.3 ready.
bird> show mem
BIRD memory usage
Routing tables:126 MB
Route attributes: 93 MB