Why can’t gravitons distinguish gravity and inertial acceleration?

Gravitons do not mediate the gravitational force and you cannot detect gravitons flashing to and fro between objects interacting gravitationally. Since you cannot detect the gravitons you cannot use said gravitons to find out whether acceleration is inertial or gravitational.

It is often said that forces are due to the exchange of virtual particles, for example the EM force is due to the exchange of virtual photons while the gravitational force is due to the exchange of virtual gravitons. But virtual particles are a computational device and do not actually exist. Those Feynman diagrams you have seen showing the exchange of a virtual particle are just a graphical representation of an integral called a propagator and do not show a physical process. I cannot emphasise this strongly enough:

Virtual particles do not exist!

Real gravitons are the quanta of gravitational waves, just as real photons are the quanta of light waves, but real gravitons do not transmit the gravitational force any more than real photons transmit the EM force.

When we write the four-acceleration of some observer we write it as a sum of the inertial and gravitational terms:

$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha{}_{\mu\nu}U^\mu U^\nu $$

where the first term on the right hand side is the inertial part and the second term is the gravitational part. However neither of the terms on the right hand side are tensors so both are changed when we change the coordinate system. It is a fundamental principle in general relativity that we cannot distinguish between the two terms since either can be made zero just by choosing appropriate coordinates. In fact this is the equivalence principle stated mathematically.

Firstly, pure General Relativity theory doesn't have gravitons, it just has spacetime curvature. Gravitons are a quantum particle, and GR isn't a quantum theory. Hopefully, some kind of Quantum Gravity theory will unite GR & quantum field theory, but we don't have a successful QG theory yet. So we don't know if gravitons even exist, but considering how successful quantum theories have been, it's not too much of a stretch to say that they probably do exist.

However, even if a QG theory says that gravitons do exist, we may not be able to detect single gravitons directly because (as Anna V mentions) they interact so weakly with other particles (and with each other). It's hard enough just detecting powerful gravitational waves. If we have trouble seeing a veritable gravity tsunami, what hope do we have of catching a single droplet from that tsunami?

On the other hand, you don't need to go looking for gravitons to distinguish between uniform acceleration and a gravitational field. The equivalence principle is local: it essentially says that a sufficiently small chunk of curved spacetime looks approximately like flat spacetime. It's not possible for a real gravitational field to be perfectly uniform everywhere. So if you're permitted to make measurements over an extended region of space you can easily detect that the field isn't uniform.

For example, if you're in a 1 km tall spaceship in deep space, accelerating perfectly smoothly at 1 g, then the equivalence principle says that it feels just the same as if you're at rest on the Earth. However, on Earth if you ascend 1 km you will measure a slightly smaller gravitational acceleration, whereas in the ship, the acceleration will be the same up in the top section as it is in the lower section.

Actually, that example may be a little misleading, since in both cases a clock in the top section runs slower than a clock in the lower section, in accordance with the equivalence principle. Alfred Centauri explains this effect nicely in this answer. This was famously tested on Earth in 1960 in the Pound-Rebka experiment. A better illustration is when we drop 2 masses. On the rocket they both fall straight down. On Earth, they both fall towards the centre of the Earth, so their paths converge slightly, as illustrated in this anim, courtesy of Markus Pössel:

curved spacetime anim

Is this what you are talking about?

In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference

Take string theories, which allow for the quantization of gravity, and have a graviton, and at the same time, can show that they are compatible with the classical (non quantized) theory of General Relativity , which is emergent, a reviw . These are two different mathematical models, each valid in its variable domains. For example thermodynamics is emergent from statistical mechanics, one does not expect or mix up laws/principles from one framework with the other. It is only conservation laws that are expected to hold for all frameworks.