One possibility is in using space based detectors. Essentially, you put the detectors up above the atmosphere in a relatively 'neutral' area of space to avoid background from other sources [0]. Then, using this floating detector, you use other sources in the universe to do the colliding for you. Something like an astrophysical jet[1] coming out of a black hole, some neutron stars colliding, some super novae collapsing, etc. Taking all these detectors together, you can then get the results you want.

The major issue is that you have nearly no idea when/where these events will occur, and they are really rare to begin with. Add the distances that these things occur at and your error bars go through the roof[2]. Still, over a VERY long time and with a LOT of these detectors spread about the Sol system (and likely a bit further out), you can get the results you'd need.

It's not cheap or easy or timely. But with 12 orders of magnitude to go, it's a lot cheaper, easier, and more timely than trying to build an accelerator with Jupiter in the way :)

[0] as in, not next to the Sun, Earth, Moon, near gravitational perturbers, etc.

[1] https://en.wikipedia.org/wiki/Astrophysical_jet

[2] http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.htm...

Building more telescopes like LIGO, IceCube and a whole bunch of other "microphones" or "telescopes" or whatever you want to clal them.

But not sure if that qualifies as "experimental" particle physics.