From what I can find, all experiments that test/validate SR use quantum sized particles. For example, the Hafele–Keating experiment used cesium clocks based on radioactive decay, particles that are subatomic. A larger test object for example might be a crystal oscillating and measuring the distance of its excursion. Could an experimental physicist give some insight as to whether it is feasible to deposit an electric charge on such a thing and accelerate it in a particle accelerator?
Atomic clocks are certainly macroscopic objects (many of them are quite heavy :)) and they rely on a variety of mechanisms to work. (Incidentally they are not typically based on radioactive decay.) If relativity somehow only affected some of these mechanisms and not others then I doubt the clock would function correctly at all.
Leaving that aside, observations verifying general relativity are typically on very macroscopic scales indeed, e.g. the precession of Mercury mentioned by @Ialala in the comments, gravitational lensing caused by galaxies, detection of gravitational waves at LIGO, observations of binary pulsars, and black hole observations.
The "strange" effects of special relativity all flow from the invariance of the speed of light, and this has been verified by a lot of experiments starting with the original Michelson-Morley one.
Interferometer experiments such as that by Sagnac demonstrate time dilation in rotating systems. The Global Positioning System (GPS) is adjusted to take into account all of the effects of relativity, including the Sagnac effect, gravitational time dilation, and velocity based time dilation.
A longer (although still incomplete) list of experimental evidence for the theory of relativity is at https://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html.
Could an experimental physicist give some insight as to whether it is feasible to deposit an electric charge on such a thing and accelerate it in a particle accelerator?
Unambiguously that specific experiment is not feasible. The "large" hadrons in the LHC are still individual atoms.
However, there are still macroscopic experiments for many aspects of SR. Also, the line between macroscopic vs microscopic is not always particularly clear. In other words, is an atomic clock macroscopic or microscopic, the clock is more than just the atoms, there are also oscillators and a resonance cavity and some form of spectrometer, all of which are macroscopic. Similarly, is a laser/maser macroscopic or microscopic, there is a "pumped" population of atoms, but there is also a resonance chamber, mirrors, etc. How about a Mossbauer rotor, it is a nuclear transition, but the sharpness depends on the fact that the nucleus is firmly planted in a macroscopic piece of metal. Stars also are very macroscopic, but we get radiation from a star that come from specific atomic transitions in addition to the overall blackbody radiation, if you can't count a star as macroscopic that seems silly, but the radiation does include quantum spectral lines.
In the end, I think such a distinction is not particularly meaningful. It is too ambiguous to form a well-motivated theoretical issue that needs to be addressed. My favorite resource for an overview of the overwhelming amount of experimental support for SR is:
All of the experiments there are valid experiments that cannot be discarded. However, I would additionally consider all of the experiments using the following techniques to be "macroscopic":
- Interferometer-based experiments (MMX etc.)
- Mossbauer rotor experiments
- Astronomical/cosmological experiments
- Electrical engineering experiments (capacitors, magnets etc.)
- Pendulum based experiments
- Cavity resonator experiments
- Experiments with transparent media
So I would consider a large number of the experiments listed to be "macroscopic".
To the above, I would also add "satellite timing experiments", where time dilation effects between orbital satellites and the ground have been measured. It's true that most of these effects are from general relativistic time dilation, but some of the effect is due to special relativity, too. Famously, relativistic effects need to be factored in order to get the meter precision that we expect from GPS.
If you read Einstein's first paper on relativity, On the Electrodynamics of Moving Bodies, his example of a relativistic effect is the fact that when you have a magnet near a conductor and move either one, the effect is the same regardless of which one you move, so long as the relative motion is the same. Relativity was the result of observations of the macroscopic world. The confirming microscopic observations came later.