I just now realized that the monkeys have this capability (reference StackOverflow), and this does open for some fun experimentation. Have you by any chance used a userscript for such a purpose before?

Depends on the level of technology you’re willing to go to:

Smart phones today you’re probably all out of luck, whether it’s Google’s Android, Huawei’s HarmonyOS, or Apple’s iOS. Same with any desktop PC.
Dumb phones without internet connectivity, like the Mudita Pure, can get you pretty far, but as you’re making calls and sending texts through your telecall provider, your provider might use that data for training purposes still.

For other technological devices, like fitness trackers, TVs, cars, fridges, or any IoT device really, that somehow either connects to internet or syncs to a device with such capability (your phone) in order to phone home, you’re likely out of luck.
Dumb devices, like mechanical timepieces, monitors, and “normal refridgerators” can get you pretty far, but remember that your purchases of these things, along with all the metadata of those purchases, might also be tracked and trained on.

I have a slight feeling that they will require “proof” of no existing social media profile: Either you give them something, anything, or they will try and search the web for any account that matches your identity.

And of course, if your cooperation is unsatisfactory, they reserve the right to deny you entry.