This is not really a solution, more so a workaround, but as I got no answers for 3 years, I decided to post it.
While I think messenger got better lately, it still compresses every media. It's no surprise - they have to store them on their own server after all. But there are a lot of messaging apps that store all media on the recipient's phone instead. Most of these send all media uncompressed, or at most compressed in a barely noticeable way, and they have the additional benefits of generally being much more private and secure than Messenger. Some I used for a longer time are:
- WhatsApp - it's very widely used everywhere in the world. Used to be renowned for its privacy, but recently there have been issues regarding that as Facebook made modifications. It sends all media without any compression.
- Signal - it's a new one, but already gained widespread usage comparable to WhatsApp. Most of the users changed from WhatsApp as Facebook made those privacy-violating steps. (In fact, the week that happened, they had so many new users they temporarily had to shut down the service as the servers couldn't keep up.) It is by far the most private one, as the whole service including the app and the server is completely open-source. Especially used in areas where privacy is more of a concern, like the USA, troubled areas like Hong Kong, or non-democratic countries, but it is gaining popularity everywhere.
- LINE - I used it on a daily basis, and among all these, it feels the most "premium". I like the interface, and the stickers and themes are, while paid, are of exceptional quality. Out of all, it's the closest to Messenger in terms of features. Mostly widespread in the Eastern-Asia region, especially in Japan. It also has an interesting backstory: originally it was created by the LINE corp. for internal usage after the Fukushima accident, to let the employees spread and gain information quickly. Then it proved useful and more and more outside people requested to join, so eventually it became public.
- Telegram - I haven't used this too much, but it is roughly on the same level as Whatsapp used to be before the disputed steps of Facebook.
- KakaoTalk - I barely know its existence, but I list it for completeness. Just as LINE, it's mostly used in Eastern Asia.