Thank you again.
You can make a shortcode of a button, and create an onclick event, when onclick, create an ajax request which will send a message using fep_send_message() function.
If it needs some code, i could give you, but it needs many code and it will be better if you or your programmer can create an extension with that code.
If you want us to create an extension for this you can hire us. https://www.shamimsplugins.com/contact-us/