Filter blockAttachmentsForm

I realize I can add this to a class file to render the form in the block editor:
$view->blockAttachmentsForm($block);

Is there a way that it could only display items having a specific class?

If it possible, would there be a way to access a parameter of said class (ie. description).

You can specify a default search filter to apply to the selectable items by passing it to the third parameter of blockAttachmentsForm. As an example, see how the Mapping module limits the selection to items with mapped locations.