No, this kind of detail is not AL compliant. When you create a character, you create that character using the rules for races (which specify typical ages and lifespans) and backgrounds from Adventurers League Content Catalogue. None of those include this particular "curse" — nor do any of them include anything which would make you quite so special. (Citation: Adventurer's League Player's Guide, page 2.)
In my experience in Adventurer's League, if you would bring this to the table, you would get one of two reactions:
- "Uh, no, that's not really a thing, sorry. Do you maybe want to be an elf? They have a long lifespan."
- "Okay, whatever." — and then this would never come up in the game, because they tend to be role-play light and combat heavy, or at least with role-play constrained within the options in the published adventure being run.
Backstories are definitely meant to be a "mechanic" in 5E. As the Player's Handbook says:
The sample backgrounds in this chapter provide both concrete benefits (features, proficiencies, and languages) and roleplaying suggestions.
These "concrete benefits" are "mechanics", and "I live forever" is certainly a concrete benefit — but a custom one, and while custom backgrounds are allowed in AL play, custom background benefits are not. You have to pick the benefit of an officially published background for your character.
That said, from the many hours I've played with different AL DMs, backgrounds usually only come up in AL in the extremely rare case when the author of the content thought to put something special for a specific background in that module. This is why you might get option 2: it's unlikely to actually matter.
I would also like to add a comment on:
To be clear, I'm not actually playing in AL, I just want to use those rules as a baseline to judge if this backstory would have any balance issues.
I don't think AL rules make a good standard for this, because AL is not concerned with general balance issues. Adventurer's League is set up for a particular type of drop-in D&D, and many of the restrictions and limitations are simply to accommodate that. To take a big example, there's no overall balance issue with mixing material from multiple sourcebooks beyond the PHB, but the "+1" rule makes it simple for game organizers and rules out a whole class of possible specific issues.
It's probably better to explain your idea directly to your DM before you get too caught up in the details of writing it up. Explain the concept and why you want it and what benefits and drawbacks you expect will come along with it. Whether your DM sees it as positive or negative — or completely neutral — will depend on the tone of the game.
For example, I ran a game and one player had the idea that their character would basically be a messiah. That didn't fit with my much more local-scale game concept and the mundane backgrounds of the other party members — my concern was that'd end up dominating the game. On the other hand, I played in a game where the DM started with "so, you are all actually all baby gods and don't know it yet", and encouraged us to come up with outlandish and miraculous backgrounds.
So, a background like this could be overwhelming in one game, and barely interesting in another. The balance doesn't depend on anything that can be assessed independently (other than to say, like above, that it certainly a bigger deal than most of the other backgrounds — compare "haunted one" for the closest kind of thing in terms of marking you as special, but note that it doesn't really change anything that's in the normal player race description).