Embed the button in the menu

I just bought this plugin, all the functions work, and with the custom CSS I was able to make it look as I wanted it to look as well.

(For me editing the Button open background color \ Button close background color \ Button open Text color \ Button close Text color attributes did nothing, but I fixed it with custom Inline CSS)

My question is, I want it to be embedded in the menu. Is there a possibility to do so?

The page I need help with: [log in to see the link]