If you're looking for a full CSS solution without registering any new bbcode, here are some ideas.
You can target all links that are in first posts .PostStream-item[data-number="1"] .Post-body a
There's no way to add class names to elements natively, either in markdown or bbcode in Flarum, but maybe you could use the title as a CSS selector:
Markdown [Link text](http://example.org 'Link title') or bbcode [URL=http://example.org title="Link title"]Link text[/URL] and select with .Post-body a[title="Link title"]
Or if the download nature of the button can be guessed from the domain/path of the URL, you could use a CSS partial selection on the link: .Post-body a[href^="https://downloaddomain.tld/file"]
