Show a different link based on IP

Yes, use a shortcode:

function aslums_first_shortcode( $atts, $content ) {
    if ( IS INSIDE NETWORK? CHECK ) {
        return 'inside the network'.$content;
    }
    return 'outside the network'.$content;
}
add_shortcode( 'aslums_shortcode', 'aslums_first_shortcode' );

In your post content:

You are: [aslums_shortcode]

As an aside, could a user outside your network bypass verification if they knew what the URL users inside the network are using? You may want to reconsider how authentication works on your network