A class in D&D isn't just a set of skills you can pick, it actually comes along with some backstory, sometimes it even requires some involvement from the world around our character (Warlock, Cleric, Sorcerers, e.t.c.). And then there is multiclassing... So... I hope I'm not too late with this suggestion. I really enjoyed how Blood Magic was handled in Dragon Age: Origins, where you could learn it only under certain plot circumstances. And I think that multiclassing in BG3 should come with something similar, being unlocked through certain actions. For example:
1. We could learn classes from our allies and companions. Halsin could help us to start our way as a druid. Volo and Gale could help us to learn arcana magic.
2. We could stumble on some guides in libraries for arcana, rogue skills, fencing techniques e.t.c.
3. We could do some quests, with ancients, devils, feys and other forces being involved, with Warlock pact provided as the reward option.
4. We could go through some clerical teachings in Temples or be granted a clerical class as the reward from some deity for doing something by their tastes.
5. And after all it could be some special Absolute cult training granting us access to whatever class we want.
I think all of it would add to roleplaying options and immerse us in our additional class better than just a button somewhere in the level up screen.
