I have recently implemented the Coderops multi level menu in my own site under bootstrap 3. My solution might not be exactly what you need as it is a Backbone Marionette App, however there might be something there to get you started.
My HAML for getting the menu DIV nested inside the bootstrap container is
!!!
%html
%head
%title>
= "Mark2"
%meta{:content => "width=device-width, initial-scale=1.0", :name => "viewport"}/
%meta{:name => "fragment", :content => '!'}
= stylesheet_link_tag "application", media: "all", "data-turbolinks-track" => false
= javascript_include_tag 'modernizr'
%body
.container
.mp-pusher#mp-pusher
%nav.mp-menu#mp-menu
.scroller#scroller
.scroller-inner
.row
.col-md-12
= yield
= requirejs_include_tag "app"
So I have the menu div inside the bootstrap container and then the rest of the site inside the scroller.
A full write up is on my personal site.
Multi Level Push Menu In Backbone Marionette And Bootstrap 3
Feel free to comment here or on my site if you want more details.