This player is using randomness to abdicate responsibility for their actions.
Using an example of something you fear happening from your post:
I don't want to be there for the fall-out when destiny finally calls for the king to be splashed with 2 gallons of water
I'd very much like to to know how the player uses dice or tarot cards to decide that throwing water on the king is the right course of action.
Except, we almost certainly already know the answer here. The player thought of a funny, but disruptive thing to do, and chose to use their preferred source of randomness "to see what destiny calls for", and give themselves narrative permission to do the disruptive thing. This randomness, be it dice roll or tarot draw, is really only the player removing responsibility by one level to disguise their intentions - being disruptive and "funny". This is true even if the player audibly, publicly, comes up with a few options, (some of which may be positive, non-disruptive, actions) and then draws or rolls to choose. This is still a player who wants to be disruptive, they're just doing what they hope will be a better job of letting the randomization take the blame. A player that didn't want to be disruptive but still wanted to let the dice decide would not have put any disruptive actions in the random list. More specifically to your problem player, the Tarot deck is infinitely interpretable - it is not providing a course of action. All the player has to do to "justify" their disruptive action is find a way to explain how it came from the card.
This is the My Guy Syndrome problem that you already know about, but hidden behind a veil of randomness. Solutions to the My Guy Syndrome are the solutions to this problem. The only wrinkle is that the player will likely try to "blame" the dice/cards for their actions. You will likely need to call them out on how this is just a dodge, and they've chosen to be disruptive. If they choose to take this dodge, and hold fast to it, it will make following the less drastic steps of the next section much harder, if not impossible. In that case, I'll direct you to the last paragraph, and the potential consequences of letting the disruptive player continue to make the sessions unpleasant.
Do you need to handle it? How should you handle it?
You said "I can assure you that they don't try to actively hamper the group's enjoyment", and yet you're here, asking for advice, because your enjoyment is hampered. If they really aren't trying to do so, then the communication should be simple: "Hey, when you do these random things that make our lives harder, I have less fun in the game. Could it be possible for you to tone it down?" You can even bring up that the random tarot stuff can be really funny, and that you sometimes get a kick out of it (if that's true, anyway) - you don't need the player to stop doing their thing entirely, just to make it less likely to cause you issues. Help the player understand when a random action is called for, and when it would be disruptive. Guide them towards doing things that are genuinely funny and do not make the other players annoyed. Importantly, make sure the player realizes that they are just one of the players at the table, and making everyone else unhappy is not a cool move. You aren't the GM, so your influence on the game is somewhat limited, but as a group of friends, your desires and enjoyment should matter to the problem player.
All that's great, but you're here, asking for advice, which makes me think the simple answer either didn't work, or you're unwilling to communicate with the player for some reason. (In my experience, this kind of hesitance is often caused by knowing that the problem player won't respond well to correction, but that's not necessarily what's going on here.)
You've said you're completely unwilling to chose the nuclear option, by which I believe you mean you're unwilling to kick the player, because you're all "best friends". Some of what I'm about to say might sound harsh, but please understand that the advice I'm giving is hard-won, from years of experience in the rpg community.
First off, a "best friend" should care about your enjoyment of the game as much as you care about theirs. That they are making your free time less enjoyable should be enough to change their behavior on it's own. Part of being someone's friend is caring about them, and their enjoyment of shared activities. Does this player, outside of the game, act in a way that shows that they care about the other members of this group?
Secondly, even the closest group of friends does not need to do everything together. If a friend is a bad match for an activity, they can (and should) skip or (in the worst case) be excluded from that activity. If you haven't, please go read Five Geek Social Fallacies, for a more eloquently put explanation of this point, and why it's not bad and wrong and being a terrible friend to not invite this player to RPG night.
If you have these conversations, and you've pondered my points, hopefully you can come to a good point with your game, and your friend. Maybe they tone it down, maybe they leave the game. But there's another, worst case option, and that's the game entirely disintegrating. Another player - maybe you, maybe someone else in the game, decides they're done with dealing with the problem player, and leaves the game. A fourth decides that without the third, they don't feel like it either. Shortly the game is ruined, because nobody was willing to make sure the game was actually fun by dealing with the problem player.