Is there even a bound on the computational power we'd need to make an advancement here?
Sounds like the LHC’s physicists have an idea.
If CERN had an unlimited budget, I suspect they'd do it however they did it before.
When I last worked there in 2015 a typical pile up situation was having about 50 collisions per detector reading. It is no simple problem to simultaneously reconstruct 50 collisions from the same set of overlapping detector measurements.
Towards the end of last year, they had to start levelling the instantaneous luminosity to 75% of what they could achieve,† primarily to reduce the load on the grid.
† Edit: the maximum peak luminosity is still 200% of the design value, so the performance is beyond initial expectations.