When on a console that allows it, save file editing as explained below is by far the fastest, as you can be done within a single in-game day. If you don't want to cheat like that, there are a few relatively fast 'legitimate' ways too:
use Jade and Crystalariums (each will produce new jade every day and 16 hours in-game), then trade the Jade on Sundays with the desert trader for stairs. Use those stairs on the luckiest type of day with luck boosts from lucky rings and food/drinks.
like this comment said, use money to buy stone, turn that stone into stairs as well. Use the same way as described in the previous point.
do what SeanyBoi5 wrote in their answer, and slay monsters in the 'dangerous' mines.
So, after trawling across the internet for a bit more, I came across this youtube video about 'luck level'. Increasing your luck level to an insane amount (like 1000) will make sure that floor 10 and every lower floor in the Skull Caverns is a Treasure Room. The youtube video lists a lot of other 'benefits' as well. Most importantly, Luck Level is something you can influence by doing save file editing, as the video doesn't disclose which mod allows the increase of luck level like it's shown there.
I ended up going with this, and found Auto-Petters on floors 40, 60, 66 and 131. I started immediately in the morning and was done at around 2PM (I did use a movement speed buff, spicy eel). If you want to, you can create extra inventory room by not bringing a pickaxe and only bring stairs to get down to floor 10, every treasure room will already have a stair. Do bring a weapon though as the treasure chambers still have enemies. Really, the weapon is the only thing you'll really need to bring.
I found a pretty good guide on generic save file editing here. The summary: In your windows file explorer, go to %appdata%, your stardew valley save files will be in de Stardew Valley folder, under 'Saves'. Each farm you have will have it's own folder, named after your character + a number. Copy the folder from the farm you'd like to edit to a save location as a backup first! You never know what you might accidentally type/delete and need the backup. It's also good to have it to look up the original value of your luckLevel once you're done and want to set it back to normal.
For each save, you need to edit 2 files: SaveGameInfo and, in my case, Tink_241984836 (My character's name). Open them in Notepad, press ctrl + f, search for 'lucklevel', and change the value between the XML tags to 1000. (So you end up with 1000). Save both files, and start the game. Happy treasure hunting!
And once you're done and want to mine the Skull Cavern as usual again, you need to change your save file once again, and set the luckLevel back to it's original number (in my case, 0). The setting persists, so if you don't, you'll have this extreme amount of 'luck' every day and will never be able to mine the Skull Cavern normally again.