This is a perfectly good idea, and there are definitely experiments that test this. The issue, however, is that there are really three possibilities. Suppose that we have a mass in superposition,
$$|\psi \rangle \sim \frac{|\text{here} \rangle + |\text{there} \rangle}{\sqrt{2}}.$$
Let the gravitational fields correspond to the two states be $|g_{\text{here}} \rangle$ and $|g_{\text{there}} \rangle$. The possibilities are:
- The gravitational field behaves like any other quantum field; it enters the superposition, so that the joint state of the field and mass are
$$|\Psi\rangle \sim \frac{|\text{here}, g_{\text{here}} \rangle + |\text{there}, g_{\text{there}} \rangle}{\sqrt{2}}.$$
- The gravitational field is inherently classical, so it has a definite value given by averaging over the possible positions of the mass, $g = (g_{\text{here}} + g_{\text{there}})/2$.
- The gravitational field is inherently classical, and it causes widely separated superpositions to collapse, through a mechanism which is beyond the standard rules of quantum mechanics. In other words, you end up with either $|\text{here} \rangle$ and $g_{\text{here}}$, or $|\text{there} \rangle$ and $g_{\text{there}}$, with 50/50 probability, not a superposition.
Using your idea, it would be very straightforward to test (2). The issue is, just about nobody believes that (2) is true! Forget about moving masses around in the lab: things have been going into quantum superpositions since the beginning of the universe, in a way that impacted cosmological structure formation. The very fact that the Earth orbits the Sun, instead of some very widely smeared out mass distribution, rules out (2).
It is possible to test (3), but it's trickier. The first issue is that the particular mechanism of collapse, and its rate, depends on the (speculative) way you extend quantum mechanics. (You recover (2) in the limit of zero collapse rate.) There is a wide range of possibilities, some of which are much harder to test than others. I don't think Penrose, who is the most famous proponent of (3), has a concrete model in mind either.
Putting that aside, you could test (2) by performing a sensitive interferometry experiment, and looking for gravitationally induced decoherence, i.e. a reduction of the interference fringe visibility. This has indeed been investigated, e.g. see Quantum Gravitational Decoherence of Matter Waves (2006) and Gravitational Decoherence of Atomic Interferometers (2002). Upcoming experiments which seek to produce macroscopic superpositions, such as MAQRO, would also automatically test this. However, it's all hampered by the lack of a concrete model to target.
The reason you don't hear about these experiments too often is that almost everyone believes (1). In general, it doesn't seem to make sense to couple totally classical and totally quantum objects, without modifying the rules of quantum mechanics. We've been through this with the electromagnetic field, where a treatment like (2) runs into paradoxes${}^1$.
As a result, the default assumption is that the gravitational field should be described at low energies just like the electromagnetic field, namely with the rules of quantum field theory. Just about all approaches to quantum gravity, such as string theory, loop quantum gravity, and asymptotic safety, try to reproduce this. It is, however, still important and interesting to check it. So it's good that a number of experiments are doing just that!
${}^1$ Sometimes (2) is used in semiclassical gravity calculations, but I think it's well-accepted that it doesn't make physical sense in general. In the cases where it is used, you don't get matter superpositions that produce radically different gravitational fields.
