PeterDonis said:
Gravitational waves are a (potential future) tool as well; the question would be whether
they are subject to the HUP. If they aren't,
something different would have to happen when they interact with, say, an electron (possibly in a different experimental setup than the "Heisenberg microscope", based on the comment by
@vanhees71), even if every other experiment we've done with electrons shows that they are subject to the HUP.
I think, what you are after is even more ambitious, i.e., you don't want to use gravitational waves (in the sense of classical ones, i.e., based on standard GR) but you even want to see quantum-gravitational effects, i.e., field quantization.
Take the analogy with em. waves: There the direct experimental confirmation of field quantization is already pretty difficult. The first indirect hint was of course already black-body radiation when treated in the kinetic approach by Einstein (1917): There you necessarily need not only the classical notions of stimulated emission (em. waves emitted by accelerated charges, where the acceleration is due to the em. field itself, e.g., the electrons/nuclei in the walls of a cavity) and absorption but also spontaneous emission, which is a generic quantum effect and due to the vacuum fluctuations of the em. field.
Now for the gravitational field it's hard to imagine how to achieve an analogue of gravitational black-body radiation or, in quantum language, a graviton gas in thermal equilibrium.
Then more direct hints at field quantization in the em. case are not so easy to find. The usual QED-tree-level results like the photoelectric effect or Compton scattering are equivalent to the semiclassical treatment, i.e., with the particles involved (here mostly electrons) treated quantum-mechanically and the em. field/waves as classical.
Since the most simple case for truly quantum effects of the em. field is the impossibility to split a photon of given frequency somehow, you need to prepare true single-photon Fock states to use quantum optical measures to demonstrate field quantization. AFAIK the first experiment on the "indivisibility of photons" is the one by Grangier, Roger, and Aspect (1986). They used an atomic cascade to have heralded single photons and demonstrated the anticorrelation effect using the heralded single photon in a beam splitter. To really see this anticorrelation, it's important that the single photon is heralded, i.e., a utmost dimmed coherent state won't do. Today of course a more convenient way is to use parametric downconversion as a heralded-single-photon source. All this is technically available for about 30 years only, and it's hard to imagine how long it will take to devolop the gravitational analogue, i.e., to produce "heralded true single-graviton states".
All these speculations of course also assume that this analogy with the electromagnetic field and its quantization somehow applies to the gravitational field too, which is not so clear. Though of course you can quantize the (free) gravitational field formally, there's still not a satisfactory quantum theory of gravitation including interactions (interactions of the gravitational field with matter as well as the self-interaction of the gravitational field, because as non-abelian gauge theory gravitons should be self-interacting at tree level if the analogy with standard-field quantization holds).
Another puzzling question is, in how far one has to take the geometrical reinterpretation of the gravitational field as spacetime geometry as is standard in the formulation of classical GR since Einstein's original forumulation seriously, i.e., in how far is spacetime itself quantized and what does this really mean on an operational and observational level.