I don't think so. You can't just "Montecarlo the hell" out of a post-quantum cryptographic scheme [1] which are also based on lattices. Even if that analogy isn't quite apt, I don't think looking for a good random lattice is a good way to find the densest possible lattice [2].

[1] https://en.wikipedia.org/wiki/Post-quantum_cryptography

[2] https://en.wikipedia.org/wiki/Random_close_pack#For_spheres