This question is indeed a little bit on the philosophical side (or perhaps this answer is!)
It is much easier (and probably scientifically more accurate) to state when a system is not BCS Cooper-paired than to say when it is. We can say that we have evidence that a material is a BCS-type superconductor, but we cannot say it is one with 100% certainty. BCS is a model and of course in any real material there will be deviations due to band-structure, electron-electron interactions, etc.
There are numerous experiments that are indicative of and consistent with the BCS theory of superconductivity. Of course, the most notable is the Hebel-Slichter peak, which BCS predicted. Then there are the Giaver tunneling experiments which showed a uniform (s-wave) gap in the density of states. There are also the phonon bumps in the second derivative of tunneling spectra analyzed in depth by McMillan that are suggestive of a phonon mechanism. Then there are the experiments with flux quantization and Josephson tunneling which show charge $2e$ quasiparticles. Of course this latter example is also present in unconventional superconductors. However, these are all suggestive of BCS-type condensation when considered as a whole.
I do believe this question is in some sense ill-posed because all of these experimental signatures, which are predicted by BCS are not necessarily specific to BCS.
Most unconventional superconductors don't conform to the BCS theory because they violate one or several prerequisites for a BCS superconductor such as:
1) Arising from a Fermi Liquid normal state
2) Being three-dimensional metals prior to undergoing the superconducting transition
3) Being adversely affected by magnetic impurities
4) Being unaffected by non-magnetic impurities
5) Being phonon-driven
6) A few others
Nature is much cleverer than us humans and it is easy to imagine her coming up with much more exotic mechanisms of superconductivity that conform to almost all but a single glaring absence of an experimental signature that we thought necessary for Cooper pairing or condensation to occur.
We should therefore not ask if a specific superconductor is a BCS superconductor but examine whether or not we can find evidence to show that it is not a BCS superconductor. If the superconductor in question keeps passing the tests, the closer we are to certainty that it is a BCS-type mechanism that is responsible for the superconductivity in the particular material.