I posed a question in the article Gyroscopes on the Moon, at r/AskPhysics and also in a dedicated r/Physics question thread but, as of the time of writing, I've not had a response.
Given how swift people have been to pile on when there is a vague hint that I might be wrong, I thought I might provide what I think might be the answer, put my neck on the line. Perhaps that might elicit a more authoritative response.
My thinking is this. The Earth is orbiting the Sun and the path the orbit describes is a geodesic. Effectively, therefore, the Earth is travelling along a "straight line" in curved space, rotating as it does so.
This rotation is where it gets a little complicated. Imagine that we had a non-rotating body which has a uniform linear motion with respect to either the galaxy or the cosmic microwave background (CMB), moving in flat space. Aim it at a large enough star so that it's just right for entering into a stable orbit. What this non-rotating body is doing is following a "straight line" in the space that is curved around the star. If I've thought it through properly, in this simple two body system there is no reason why the non-rotating body should start rotating. However, the curvature of space will be such that, once in stable orbit, the body will show the same face to the star at all times – it will, in effect, be tidally locked. To an observer of the system, it will appear as if the body has a rotational period equal to its orbital period – but, in another sense, it's not rotating at all. It makes sense that orbiting bodies tend towards being tidally locked because this is a minimum energy state, the relativistic equivalent of being stationary.
A gyroscope on a non-rotating body with uniform linear motion does not undergo precession. This should not change once the body is captured in an orbit – being in a stable orbit will put it in free fall, so not even the gravity of the star will be a factor.
So, how can we relate this situation to the Earth?
The Earth is certainly in a stable(-ish) orbit around the Sun, so the orbit itself should not have any effect on gyroscopes on Earth.
However, the Earth is not tidally locked, and it therefore rotates rather than being non-rotating, so gyroscopes on Earth will be affected by that rotation. However, a gyroscope on the Moon, which is a tidally locked body, should not be affected by the Moon's apparent rotation. (Note that this is merely a conclusion I've come to, not a statement of fact. There may be errors in my thinking, something that I've overlooked so that lunar gyrocompasses would actually work. I've not been able to see any indication that this has been tested. The lunar rover had a "navigational gyroscope orientated in relation to the Sun") rather than a gyrocompass per se. There doesn't appear to be any intention on relying on gyrocompasses in plans for the next lunar mission either.)
There are a couple of apparent flies in my ointment. The first is that the Earth is on a tilt with respect to its orbit of the Sun (from the ecliptic), and this tilt does not appear to change during that orbit. At perihelion, when the Earth is closest to the Sun, in early January, the North Hemisphere is tilted away from the Sun but at aphelion, when the Earth is furthest from the Sun, in July, the Northern Hemisphere is tilted towards the sun.
If we conceptually smoothed out space, so that the Earth was following a straight path, this would mean that its axis of rotation would be moving in an annual cycle (and we've just removed that cycle by turning the orbit into a conceptual straight path). I did consider, momentarily, that it could have something to do with the fact that the Earth's orbit is elliptical and the direction and extent of the tilt is related to the eccentricity of the orbit. This is highly unlikely though because the tilt is 23.5° from the ecliptic while the eccentricity of our orbit is minuscule at about 0.0167, making it pretty much circular even though even this tiny eccentricity does have a minor effect on our seasons (warmer summers in the south, which is counteracted by the greater proportion of ocean to landmass in the Southern Hemisphere which actually makes it cooler overall than the Northern Hemisphere). It's more likely, I thought, that there's another factor at play.
The other apparent fly in the ointment is that the Earth itself precesses. There's a 26,000 year cycle in which the tilt of the Earth changes, such that while Polaris is currently the (north) Pole Star, that won't be the case in a few thousand years. Vega will apparently take on the role in about 13,000 years. This precession manifests because the Earth itself is acting like a gyroscope and some force is making its axis of rotation tend to point towards the north ecliptic pole (also see this graphic). Note that this means that, on average, over a period of 26,000 years, the axis of the Earth's rotation aligns with the axis of the Earth's orbit of the Sun (axis of the ecliptic).
If the Earth itself is being precessed by its orbit around the sun, with a period of 26,000 years, then a gyroscope should also be precessed by the same phenomenon, but probably only very weakly. I'm not quite sure what the problem is with my geodesic thinking, but it seems to be wrong and therefore, I would have to presume that gyrocompasses would work on the Moon. (It'd still be interesting to see the results of an experiment though.)
As to whether there would be a wobble or an offset, I think that (for a gyroscope on Earth submerged in a viscous fluid so as to act as a gyrocompass) there would be a very tiny offset from the north celestial pole towards the north ecliptic pole and this offset would change over a period of 26,000 years – so yes, I think there would also be a wobble, in that offset, or perhaps more of a circle, but either would be pretty hard to detect over such short periods as a human lifetime.
Of course, I might be wrong, I might just be telling myself a complicated story using concepts that don't cohere as well as they might first appear to, but please don't let my free admission stop you from attacking the ideas above if you think they are misled.