Here are a variety of options, depending on where you want to go with this exactly.
1. A self-contained joke or prank
In my experience, the best way to do this is not to make it part of the actual game. Treat your joke like a more elaborate version of your normal tabletalk &c.: it's part of the social experience of your game but doesn't intrude on the fiction or the mechanics.
For example, one time, playing D&D, the players got to a big encounter, and I started an elaborate description that got sillier and sillier while setting up a bunch of minis I'd prepped (I was using paper minis) of bad-guy monsters from children's cartoons. Then they got the joke, laughed a bit or groaned at the dumb puns, and I put all that stuff away and narrated the scene from scratch "for real."
The result was a self-contained moment of amusement that proved moderately memorable but didn't disrupt the game any more than a normal days' goofing-around or arguing about dinner plans would have. That's about the desired effect for a one-off joke. Simple, easy, non-disruptive.
2. A purposefully goofy session, also self-contained
If you want to spend all or most of a session in a purposefully goofy mindset, I suggest running a different game for the night (or part of the night). Play Roll for Shoes or Toon or something. The change of system encourages players to see it as its own thing. Try to make it feel a little bit disposable to encourage everyone to cut loose and play for big stakes, so you can get a satisfying story in a short timeframe.
If you really want to tie it into your existing campaign somehow, set it up as a work of fiction in-game. I don't recommend the "dream sequence" because that's likely to make you feel like you're making a mockery of a protagonists' thoughts and feelings. (My rule of thumb: play dreams when you really want to go deep and weird, not shallow and fun.) No, make it something like a puppet show or a mummers' play — something that's recognizably a stylized, comical form. One appeal of this method is you can get the slapstick out of your system but also mix in subtler humor in the form of ironic ignorance (retelling a story you've played, but the public version's all wrong). You could also use it as a vehicle for accidental foreshadowing — put some weird stuff out there and see what feels interesting, but don't treat it like an obligation to force it to be true later.
3. Full integration as a thematic element
Okay, but what if you really want to both play your regular game and incorporate the holiday (April Fools, Halloween, whatever)? And yet you're worried it'll mess up your tone and theme?
There's an easy answer here: let your campaign's tone and theme mess up the holiday instead.
Have the background characters of your world celebrate their little holiday (peasants love festivals; nobles and wealthy burghers love parties; monks have a rather suspiciously opulent feast day schedule they observe like clockwork), then put your bloody hand prints all over it like you do with everything else. How many times does your source material show the audience a party that goes really, really bad, after all?
See Falc's answer for some more detailed suggestions on how to run it as a background event in a serious story.
My recommendation for you, specifically
(Based on your specific parameters.)
If you think you can manage the context-shifting involved, start with #2 and pivot to #3. Cast the players as characters/actors in a wild and fanciful story-within-a-story that represents a farcical and muddled retelling of some bit of important history (including, ideally, your PCs' own deeds). Then, when you reach a natural stopping point or the gimmick starts to get tedious, switch back to your "real" campaign and introduce some new twist or complication that resonates with the symbolic story you just played out.
Use a super-simple alternative system — something just above the complexity of coin flips, mostly to add unexpected outcomes and encourage a bit of character differentiation — so that players know they don't have to think strategically or worry about long-term consequences. Roll for Shoes should work nicely.
If you pull it off right, it'll be a big and memorable thing that feels just like one of those TV episodes that suddenly put you in a minor characters' shoes to get a different perspective on the action for a while &mdash. And if the experiment misfires, you can wrap up it early and push on to your "real" session without too much disruption to the campaign as a whole.