As always..."there's a plugin for that"...https://wordpress.org/plugins/custom-ribbon-maker/ This looks like some css that would do the trick though it might be limited on ie8 https://css-tricks.com/snippets/css/corner-ribbon/ and http://www.bypeople.com/css-ribbon/ looks pretty.
If you want to do it yourself, if you have a logo image, the easiest way would be to edit the logo image and put the ribbon on that.
You could edit the logo link in your WordPress settings to go to a feedback page and use something like the ContactForm7 plugin to handle the feedback. That might be the tidiest way to do it and it would avoid messing with templates.
If you want to get into editing templates, you could put it into header.php in your child theme if you have one ( I think you do as you mention custom css ) - or in the main theme's header.php but that is not regarded as good practice (editing theme files).
If you don't have a child theme there is a good WordPress.org page on that topic and it really is worth learning to do it all that way, as you never need to worry about losing your edits when your theme gets updated see:https://codex.wordpress.org/Child_Themes
Your theme folder would be wp-content/themes/yourthemename
and you would put the child theme folder next to it. You would then put your edited header.php in there alongside the css file which is what tells WordPress you have a child theme.
In header.php in your theme folder you should find something like:
<h1 class="site-title"><a href="<?php echo esc_url( home_url( '/' ) ); ?>" rel="home"><?php bloginfo( 'name' ); ?></a></h1>
You could add your Beta Ribbon image next to that as an <img
tag. This custom example assumes you put a jpeg called blueribbon.jpg
into your theme's images folder and have a feedback page called feedback.php. You would need to add your own class names or id to style it and you could place it in front with a z-index:
<a class="logo-section" href='<?php echo esc_url( home_url( '/feedback.php' ) ); ?>'<img src='<?php echo esc_url( get_bloginfo('template_directory').'/images/blueribbon.jpg' ); ?>' class="logo" alt='Blue Ribbon'></a>
Your logo image is in a piece of script like this:
<img src='<?php echo esc_url( get_theme_mod( 'logotype_url', $defaults['logotype_url'] ) ); ?>' class="logo" alt='<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>'>
</a>