I also use this plugin, and here’s a suggestion…
The class for that button is ‘wpas-btn wpas-btn-default wpas-link-logout’, so that’s what you’d need to target (probably with ‘display:none’).
The article at https://getawesomesupport.com/documentation/awesome-support/templating/ explains how to add a css file which won’t be over-written when the plugin updates.
It also explains how to modify various elements, and if you add a copy of ‘ticket-navigation.php’ to the suggested folder it’s easy to either remove that login button or change the page to which it directs… in my install I’ve created a simple WP page with a ‘you’re now logged out message’.
The appropriate code is:
<?php wpas_make_button( __( 'Log-out', 'awesome-support' ), array( 'type' => 'link', 'link' => wp_logout_url(), 'class' => 'wpas-btn wpas-btn-default wpas-link-logout' ) ); ?>
And wp_logout_url() can be modified to: wp_logout_url( ‘add_your-desired-url_here ‘ ).