When you ask for a "perfect" or "true" inertial reference frame you are asking for something that cannot be answered in physics. Perfection is only possible in mathematics, not physics.
So in physics, what can be asked is whether or not a given reference frame is an inertial frame to a certain level of accuracy. The surface of the earth is not an inertial frame because of the gravitational field of the earth - not because the earth is moving around the sun and the sun is moving around the galaxy. But if you consider motion only in a horizontal plane on the surface of the earth and if you are only doing the typical high school physics tabletop experiments, the earth is an inertial reference frame as far as the accuracy of the measurements performed is concerned. If you do more accurate measurements, then it would not be an acceptable inertial reference frame.
Consider a satellite in orbit around the earth and examine a relatively small volume near the center of mass of the satellite. That small volume over a suitably small period of time will be an inertial reference frame to a very high level of accuracy. For example, two small masses that are 1 inch apart (radially) in orbit around the earth that start out "exactly" at rest relative to each other will over a time period of 10 seconds come to have a relative speed of 0.006 inches/second due to the differences in orbital velocity for two orbits that differ by 1 inch. So it depends on the level of accuracy needed for an experiment that you want to perform in an inertial reference frame.
To get a reference frame that is more accurately inertial it would necessary to be orbiting much further from all gravitating objects. Thus, it is all about the level of accuracy you require of the inertial reference frame.
Your question will eventually lead you to Mach's Principle. It is an old, yet unsolved question, that still remains at the stage of "philosophical idea".
I understand that your question is equivalent to "What would be found if we could measure all effects on the pendulum with infinite accuracy?", what if even the tiniest contributions could be registered? (Please read the note at the end as well, regarding the effect on any pendulum of the proximity of mass, whether that pendulum is in a free-fall orbit or not. The effect of earth's orbital motion is not zero because it affects the speed rate of proper time)
Yes, some components of the acceleration on the pendulum allow to deduce that the pendulum belongs to a rotating frame. That leads to think that the pendulum and the whole Universe may eventually be found to be rotating around some point, but that idea makes no sense (what is that point then, if everything is rotating? Rotation relative to what?). Then Mach's principle comes to the rescue, telling us that inertia effects on your pendulum arise somehow from the influence of all the other objects of the Universe, from here to the most distant ones. But there is no mathematical model for such thing, not even in General Relativity.
The pendulum is blindly affected by the local conditions of space and time, which constantly change in time and from one point to another (although all effect other than those arising from the rotating frame on top of the bulk mass of the Earth are extremely tiny). Those conditions are determined by the arrangement of energy/mass and momentum around. In the newtonian model, by the mass distribution. This is useful because you can idealize a portion of the Universe in a model that allows you to predict some behaviour of the system: for instance the Schwarzschild metrics allow to accurately synchronize the clocks of the GPS satellites in their motion around the Earth, and to accurately model orbits close to the Sun. The homogeneous and isotropic Universe model allows to derive properties of the expansion in the past, etc. But there is no model for an accurate description of how the whole universe is affecting your pendulum.
In other words, the essential origin of inertia is still unknown. What is a Foucault pendulum eventually rotating around? There is no answer to that question. Moreover, it is not yet clear whether the question makes sense or not.
The most close answer to your question may be found in our motion relative to the Background radiation, found by means of the dipole anisotropy of the CBR. This is the closest thing that there is, to an "absolute reference frame" but it makes sense only for us. Other distant observers in out expanding Universe will have a completely different perception.
EDIT:
As correctly stated by Ben Crowell, the orbital motion is a free fall, and therefore its dynamical effects on the pendulum are different from those of being on top of the rotating Earth. However, that free fall happens along places with different values of the gravitational potential (bigger in January, for instance) and therefore the speed rate of pendulums is affected. Thus, your pendulum, as any other clock-alike device, is affected by all the other masses in the Universe.
You might think about placing several synchronized pendulums at different distant points on the surface of the Earth and, by measuring (with infinite accuracy) their speed rate differences, map some properties of the gravitational potential in which you are embedded, deducing for example the direction of a center of mass. This makes an interesting question if you want to start another post.
As for Mach's principle, let me stress that it is merely a philosophical idea, that may or may not some day lead to a real theory. It is neither correct nor incorrect.
There is often a fallacy motivated by the Equivalence Principle, in which people ignore the different speed rate of proper time inside the free-falling elevator. Yes, the man inside the free-falling elevator is unable to distinguish if he is in a gravitational field (but in free fall), or if he is floating in interstellar space, far away from any mass. But in the second case, the man inside the elevator is ageing faster that the one that is in free fall (orbit) around the Sun. This is another kind of twins paradox that is often forgotten.
Best Answer
Let's look at the definition:
Italics mine.
The crucial word is conceptually. It carries after it the whole concept of measurement, and physics is about experiments and measurements, and the theories and definitions are tools to describe and then mathematically model the observations, so that one gets a predictive theory.
Measurements come with experimental errors, and thus how complicated the theoretical model one is using depends on these errors. One takes the simplest assumptions, it makes no sense to use the galactic reference frame ( we are also rotating around the galactic center) when measuring a force on bodies on earth, and also the measurement will depend on our measuring instruments.
For example, for usual engineering uses we accept that the earth is flat, the errors of the curvature of the earth to the details of a building are so small that they are within measurement errors.
We accept that the earth is rotating, a non inertial frame, when calculating the coriolis force and the distances planes travel etc, because there the force from the rotation effect is larger than the instrument errors.
So it depends on what you are measuring, whether you can use/assume that the earth is in an inertial frame within measurement errors or not. It depends on the problem at hand.