Since the small viewport navigation ends at 767px and the menu breakpoint is 768px, let's copy the styles out of the un-packed Bootstrap CSS and use the same or greater specifity -- if possible, they use a good bit of !important (bummer) -- and put the CSS in a min-width.
HTML: Use accurate implementation of navbar construction as per the examples on GetBootstrap.com
There is no such class .navbar-inner in Bootstrap 3. Look at the migration guide (http://getbootstrap.com/migration/) and the examples on GetBootstrap.com
CSS:
body { padding-top: 70px }
@media (min-width:768px) { 
    .navbar-header { float: none }
    .navbar-left,
    .navbar-right { float: none !important }
    .navbar-toggle { display: block }
    .navbar-collapse {
        border-top: 1px solid transparent;
        box-shadow: inset 0 1px 0 rgba(255,255,255,0.1);
        overflow: hidden;
    }
    .navbar-fixed-top {
        top: 0;
        border-width: 0 0 1px;
    }
    .navbar-collapse.collapse {
        display: none!important;
        visibility: hidden!important;
        height: 0px!important;
    }
    .navbar-nav {
        float: none!important;
        margin-top: 7.5px;
    }
    .navbar-nav>li { float: none }
    .navbar-nav>li>a {
        padding-top: 10px;
        padding-bottom: 10px;
    }
    .navbar-collapse.in {
        -webkit-transition-timing-function: ease;
        -o-transition-timing-function: ease;
        transition-timing-function: ease;
        -webkit-transition-duration: .35s;
        -o-transition-duration: .35s;
        transition-duration: .35s;
        -webkit-transition-property: height, visibility;
        -o-transition-property: height, visibility;
        transition-property: height, visibility;
        display: block !important;
        height: auto !important;
        padding-bottom: 0;
        overflow: visible !important;
        visibility: visible !important;
    }
    .navbar-nav .open .dropdown-menu {
        position: static;
        float: none;
        width: auto;
        margin-top: 0;
        background-color: transparent;
        border: 0;
        -webkit-box-shadow: none;
        box-shadow: none;
    }
    .navbar-nav .open .dropdown-menu > li > a,
    .navbar-nav .open .dropdown-menu .dropdown-header { padding: 5px 15px 5px 25px }
    .navbar-nav .open .dropdown-menu > li > a { line-height: 20px }
    .navbar-nav .open .dropdown-menu > li > a:hover,
    .navbar-nav .open .dropdown-menu > li > a:focus { background-image: none }
    .navbar-default .navbar-nav .open .dropdown-menu > li > a { color: #777 }
    .navbar-default .navbar-nav .open .dropdown-menu > li > a:hover,
    .navbar-default .navbar-nav .open .dropdown-menu > li > a:focus {
        color: #333;
        background-color: transparent;
    }
    .navbar-default .navbar-nav .open .dropdown-menu > .active > a,
    .navbar-default .navbar-nav .open .dropdown-menu > .active > a:hover,
    .navbar-default .navbar-nav .open .dropdown-menu > .active > a:focus {
        color: #555;
        background-color: #e7e7e7;
    }
    .navbar-default .navbar-nav .open .dropdown-menu > .disabled > a,
    .navbar-default .navbar-nav .open .dropdown-menu > .disabled > a:hover,
    .navbar-default .navbar-nav .open .dropdown-menu > .disabled > a:focus {
        color: #ccc;
        background-color: transparent;
    }
    .navbar-inverse .navbar-nav .open .dropdown-menu > .dropdown-header { border-color: #080808 }
    .navbar-inverse .navbar-nav .open .dropdown-menu .divider { background-color: #080808 }
    .navbar-inverse .navbar-nav .open .dropdown-menu > li > a { color: #9d9d9d }
    .navbar-inverse .navbar-nav .open .dropdown-menu > li > a:hover,
    .navbar-inverse .navbar-nav .open .dropdown-menu > li > a:focus {
        color: #fff;
        background-color: transparent;
    }
    .navbar-inverse .navbar-nav .open .dropdown-menu > .active > a,
    .navbar-inverse .navbar-nav .open .dropdown-menu > .active > a:hover,
    .navbar-inverse .navbar-nav .open .dropdown-menu > .active > a:focus {
        color: #fff;
        background-color: #080808;
    }
    .navbar-inverse .navbar-nav .open .dropdown-menu > .disabled > a,
    .navbar-inverse .navbar-nav .open .dropdown-menu > .disabled > a:hover,
    .navbar-inverse .navbar-nav .open .dropdown-menu > .disabled > a:focus {
        color: #444;
        background-color: transparent;
    }
}