Unfortunately, all the methods I am trying, such as putting the embed text in a widget or going through and adding code to functions.php to remove the script tag.
I think this is an implementation issue and not something BPS is blocking. Embed code would normally go in a Header file since you need it to load when the Header loads. Try using the WordPress Code Embed plugin to add a BlueSky Feed to your site. See below.
Method 3: Using a Plugin (for Advanced Functionality or Scripts)
For embedding more complex scripts (like JavaScript for tracking) or managing many code snippets across your site, plugins like WPCode or Code Embed offer more control and a safer way to manage code.