You are using the Menu module and trying to get this effect in your mobile menu. You already spent lot of hours and stuck or frustrated to do it. Here is the simple solution for you. Make sure that you already installed the Beaver Builder plugin and created the site menu using BB menu module. Now follow the instruction at below:
Setting Responsive Toggle Type
Click on “rench” icon of your menu module and select the “Hamburger” or “Hamburger Icon+Label” option from Responsive Toggle drop down list.