How do you know which organisms are keystone species?

According to the wikipedia page, a keystone species is:

A keystone species is a species that has a disproportionately large effect on its environment relative to its abundance.

Thus, you could work out the keystone species in an ecosystem by sampling population size and and what size 'effect' the removal of the species from the ecosystem would have. If you standardised the units of 'effect' for all species in the ecosytem, the keystone species would be those with the highest effect/population fraction. I don't know if there's a critical size of this fraction for a species to be determined 'keystone', but if there is it would allow you to explicitly define a keystone species. This would rule out the problem of interdependency as you mention.

(Posting different answer as it gives a different approach to the question)

Re-reading this I think you were asking about how we'd infer these keystone species experimentally.

As a starting point, one could apply network inference on species counts in the ecosystem (https://doi.org/10.1890/09-0731.1, https://doi.org/10.1038/s41598-017-07009-x, https://doi.org/10.1016/j.ecolind.2015.11.031, as examples). This kind of study returns networks which represent the interactions between species - you could then search through these networks for species with many connections in the network. Removal of a species with many interactions with many other species is likely to have a greater impact than removal of a species with fewer interactions, suggesting these are keystone species.

I realise in your question that you sort of imply each species has an interaction with every other species in the ecosystem, but studies like the ones above show that this might not be true. Or at least, many of these interactions may be indirect, via another species.

After such a study you could apply statistical methods like Pearson's Correlation to work out the 'strength' of the interactions in the network. From this could identify species with many high-strength interactions as keystones. Alternatively you could attempt a parameter fitting on the learned network structure (as you can in the case of Bayesian Networks above), to try and predict what happens to the abundances of other species if you remove one of your candidate 'keystones'.

Hope that gives a more experimental angle on how to solve this question!