I am an active player and GM in Adventurer's League. I rarely get to play CoC, I am the lone Keeper in my area at present. I think league play of some kind is a good idea.
CoC is far more deadly to characters than D&D, the potential for a character to not make it from game to game is much higher and without access to reincarnation or other magical intervention this is an important consideration. Don't have a good answer for that yet.
Adventurer's League has a tracking system and specifically designed Adventure League scenarios. So a character will log a specific adventure and can't play that adventure again with the same character. As a parallel the Scenarios in Berlin the Wicked City would be "legal", but scenarios developed by a Keeper using the sourcebook, say using an episode of Babylon Berlin as inspiration, would not. As an example for CoC, I ran the Necropolis (event scenario) for a group, each character would have a log sheet noting that they survived the adventure and then not be able to use that character for another session of the same scenario.
Consider a reward for the session, in D&D it's gold and magic items and a level. For CoC maybe borrow a skill from BRP like psionics and give a player an ability like telepathy. Or allow the character to choose to advance one of the used abilities instead of rolling for it. Maybe even a "useful" sanity quirk like "feel no pain", or "fearless".
I agree with the statement above that a character should stay in a timeline. However, it could be possible to age a character to let them play say a 1920's investigator is aged 20 years to play in a 1940's scenario.