Everything in the DLC is obtainable before beating the game except catching Calyrex.
According to this Reddit thread, you can get a Kubfu and evolve it into Urshifu without beating the game. Since trainers on Isle of Armor have lower levels before becoming Champion, this is actually a pretty easy task.
The Crown Tundra, however, does not have level adjustment. Fortunately, the very first battle with Peon does not require you to win to progress. This means the entire level 60 Crown Tundra is available to you from the beginning. You can catch as many Pokemon in Dynamax Adventures, (including Legendaries) that you want. You can't catch wild Pokemon until you meet the necessary gym badge requirements though.
Like mentioned before, you can play through the entirety of the Crown Tundra if your team can handle it, but when encountering Calyrex Shadow/Ice Rider, the game will tell you to come back once you've become Champion. (Source)
Please note I could not find whether or not the Galarian Birds are obtainable prior to beating the game, so be advised that may not be possible.
Hope this helps!