Rules as written: shrugs.
This is a great question, and exactly the kind of question to ask the DM before you try it. The guidance on "what is a game statistic" is pretty clear for monsters, and much less clear for player characters. In this answer, I outlined what I believe is the full extent of the guidance available for what counts as game statistics for player characters:
Game statistics here refers to everything mechanical about your character. That is, all of your racial traits, class features, ability scores, everything. We see this explained for NPCs in the Monster Manual:
A monster’s statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.
But this applies to PCs as well - your statistics are everything that make you, well, you, within the calculus of the game rules. In the Player’s Handbook, we have reference to game statistics in the step by step character creation section:
Your character is a combination of game statistics, roleplaying hooks, and your imagination.
There isn't much to go off of here. We simply do not have adequate guidance to make a ruling with solid rules support. In this answer, Rubiksmoose outlines all of the guidance for NPC/Monster statistics, of which there is quite a bit. But, the section Rubiks provides for PCs is quite short:
The statistics for a PC include all of the same things as for monsters and NPCs. The description of building an NPC like a PC (above) confirms that creatures built this way count those things as statistics and really that just makes sense.
Basically, anything on the standard PC character sheet is considered to be a statistic.
I think this is probably the right ruling, and it does a good job of telling what is included, but an definitive method of determining what is not included is not given, as far as I can tell.
Reductio ad absurdum: working backwards to a ruling from what is possible if we permit this.
However, I think we can arrive at a satisfying ruling by considering the consequences of allowing the Fates card's effect to be copied to a simulacrum. We know that rules as written, we can create an arbitrarily large army of simulacra given the time to do so. Ergo, if we allow the Fates card's effect to propagate to an army of simulacra, we would be able to erase an arbitrarily large number of events from history.
As a DM making a ruling, ask yourself: is this kind of reality breaking possible in my universe? If not, then the Fates card's effect doesn't transfer to simulacra.
In the end, as with all world changing shenanigans, the players and the DM should work out together how they shape the world.