Is there such thing as curvature uncertainty?

Good question. That's no surprise. We still have not reconciled GR (general relativity) with quantum theory (QT). We don't know how to do what you ask.

The measurement of a particle's position can be done exactly in QT, as you said, and it means that you've picked a position eignestate of it. The momentum variable then has infinite variance, in that state, thus confirming the uncertainty principle. This is true for a quantum particle in a FIXED spacetime background, i.e., in a specific curvature. We know this because we have worked out how to deal with quantum particles or quantum fileds Ina fixed background.

But you then go a step further, as you should, and say, well, that momentum, then must make the source of the gravitational field a quantum entity, and the field then must be quantum, and thus some aspect of it must be in a quantum state. If it's a small effect, we can treat it perturbatively, and calculate that effect. But then there are perturbations again on the position, and so on, where we have to do an infinite number of perturbations. We don't know how to calculate that without the infinities it involves; quantum gravity, done that way is not renormalizable, it's been proven to be so. If you wanted the first perturbation, and it was small, like one particle changed, it would actually converge easily, but in fact all particles, all sources of the gravitational filed enter in, and we have to account for all of them. Because everything couples to a gravitational field that's been a problem.

The good news is that gravity couples weakly to matter and anything else, and we can do a 1 step approximation and in fields which are not too strong get close enough to an answer. In fact, mostly we simply treat the particles affected as very few, and treat them quantically, and treat the source of the gravitational field (I.e, curvature) classically. That's what was done by Hawking and others in treating how a quantum effect causes radiation form a Black Hole.

The current work on quantum gravity goes along multiple lines of research, with string theory and loop quantum gravity being two of the most popular. An equivalence between gravity in a (specific, AntideSiter) spacetime and a conformal QT in its boundaries is another approach, called AdS/CFT correspondence. Till we figure that out we have no answer for your question.

The popular conceptual view of what gravity may look in the quantum realm, which would make itself manifest at the Planck scales of $10^{-33}$ cms is a bubbling and rapidly changing foam that represents the precursor to spacetime at larger normal distances.

Google Quantum Gravity and see as an intro the wiki article at https://en.m.wikipedia.org/wiki/Quantum_gravity

Read chapter 21 in Misner-Thorne-Wheelers Gravitation. In it you will see the derivation of the Arnowitt-Deser-Misner (ADM) approach to relativity. I will not reproduce that here, but sketch a bit of this. The Riemann curvature results in a Hamiltonian constraint $H~=~G_{ijkl}\pi^{ij}\pi^{kl}$. Here $\pi_{ij}$ is the momentum metric conjugate to the metric $g_{ij}$ for a spatial surface and $G_{ijkl}$ is a superspace metric. We have then the natural quantization condition $$ \hat\pi^{ij}~=~-i\frac{\delta}{g_{ij}}, $$ where it is clear there is the commutator $$ [\hat\pi^{ij},~g_{kl}]~=~-i\delta^i_k\delta^j_l. $$ This leads to an uncertainty relationship between the momentum metric operator and the spatial metric. The uncertainty relationship can be shown in a standard way.

The momentum metric is constructed from the extrinsic curvature, and so this is a form of uncertainty between curvature and the metric. This can be carried further with the Hamiltonian constraint. A good candidate to look at would be a product of metric components $g_{ij}g_{kl}$.