Louis snatches anything that comes through the letterbox and I'm sure that would include fingers. Because of the layout of our hall I can't fit one of those wire cage things over the letterbox, so I have a babygate across the hall so that neither dog has access to the letterbox or door.
I think if the man makes an official complaint it's possible you may be contacted by the police or dog warden, so if you can have something fitted to the letterbox or an outside letterbox fitted asap, you can show that you've already taken action to stop something similar happening again.
Looking at it from the delivery man's point of view, he had no reason to think that he might get bitten, so I expect he was shocked as well as in pain when it happened and I'm sure his instinctive reaction was to pull his hand back, which could have made the injury worse. But what else could he do?
I'm sure that most of us dog owners would think twice about putting our fingers through a letterbox with a dog the other side of it, but people who don't have dogs might not realise. If this man is employed by a delivery company he may have had some kind of safety briefing or guide from his employer. So if he makes a complaint that might be something you could follow up. But if he's self employed, as many of these drivers are, he might have no idea that some dogs snatch things that come through the letterbox, and he might lose income if he can't drive or handle parcels for a few days, which is another reason he might make a complaint.
Hopefully nothing will come of it and the man will have got over the shock of it by now and will feel calmer and will just make a mental note not to do the same thing again. Even if it's followed up I really don't see that your dogs will be labelled as dangerous. But taking steps to ensure it doesn't happen again is sensible just in case it's followed up.