When I review the outline of Lamrim (which I understand is recognized by the Gelupka, Kagyu and Nyigma schools), I do not see any mention of Mahamudra (or Dzogchen for that matter).
Lam Rim is Mahayana practice. How quickly they get into tantra & Mahamudra depends on the school. Gelugpas have extensive academic and Paramitayana practices (such as Shantideva), then tantra and Mahamudra. Kagyus get into tantra quite quickly, relatively speaking, and then Mahamudra. Nyingma have a very abbreviated Lam Rim and get into tantra & Dzogchen.
Does it fall into the "Emptiness Section" in the "Great Scope"?
Yes, they are emptiness practices.
Those are overly simplistic answers to you question, and as such not 100% accurate, but it is close enough to give you some perspective.