How can I replace content in the WP Admin area before an admin page is rendered?

There’s no filter for filtering all of the HTML output in the admin. The closest thing is the gettext filter:

function wpse_275812_filter_text( $translated_text, $text, $domain ) {
    $translate_text = str_ireplace( 'foo', 'bar', $translated_text );

    return $translated_text;
}
add_filter( 'gettext', 'wpse_275812_filter_text' );

This filters any text output by the internationalization functions (__(), _e(), etc.). So it will only work if the plugin you’re trying to filter uses these functions to output its text.

If it doesn’t, there’s nothing you can really do.