Firstly a minor frame challenge. I'm assuming that the premise of this question is to get it such that human civilisation has to suddenly deal with the fact that it is in a simulated universe. The problem with this is that major paradigm shifts in human understanding, even when based on the strongest possible scientific evidence, take a long time to become accepted by the scientific community and longer still by everyone else. Theories that we currently regard as having overwhelming evidence to support (e.g. Relativity, Quantum Mechanics, Natural Selection, etc.) took decades or in some cases centuries to reach widespread acceptance. All that time, Scientists were debating them and trying show where they failed. These theories are accepted now because decades of scientific research have failed to deal a serious blow to them, and because they have useful explanatory power. If you want everyone to accept the simulation hypothesis, you need to accomplish 3 things:
1. Find a series of anomalies in current understanding that cannot be explained by the existing theories, but can be explained by some simulation hypothesis.
2. Demonstrate that the simulation hypothesis has useful explanatory power based upon some particular hypothesis about the underlying workings of the simulation.
3. Give scientists and society in general plenty of time to work through it's objections, and eventually accept a paradigm shift.
With that in mind I would propose that you need to rely on a mistake/laziness of the simulators. Therefore I propose the starting point of your paradigm shift is:
The Cosmic Censorship Hypothesis
Black Hole singularities are objects hypothesised to exist, but which are fundamentally unresolved in physical models. For a start, they requires such weird concepts as infinite density. 781
Whoever wrote the model that our simulated reality runs on realised that the mathematical models they used to simulate most of the universe break down at the point of a singularity, throwing all kinds of error messages and requiring the simulation to halt. So they designed the simulation to ensure that any singularity is always cloaked in a barrier that prevents the singularity interacting with the rest of the universe - an Event Horizon.
However, far too late in the development cycle, they realised that if a collapsing star is rotating too rapidly when it forms a black hole, it can lead to a distorted toroidal event horizon which exposes the singularity to the outside universe. This is known as a "naked singularity".
It turns out this would cause all those errors to be thrown again, so they did what any self respecting programmer on a deadline would do, and they bodged a quick fix. If a naked singularity ever begins to form, a patch of space a few lightyears wide resets to an early saved state with a hard cap on maximum rotation.
Your researcher doesn't know this, but they are confused when studying older radio-telescope data of a fast rotating star that was thought to be collapsing into a black hole. They note that the signals seem to repeat, but the doppler shift distribution appears to suddenly narrow for no physically explicable reason. The whole thing just resets, but slower.
The observation can be repeated on other stellar collapse events. Once astrophysicists know what they are looking for, it becomes something of a tell-tale. Systems with more time-sensitivity are built, showing that the reset is a truly discontinuous process.
At this point, physics is in crisis and the Simulation Hypothesis is merely one candidate among many to replace conventional cosmological models. However, it starts nudging it's way up the field when other theorists are able to demonstrate that some other unsolved problems in physics are down to obvious (in hindsight) bodges and limitations by the simulators. Someone demonstrates that a workaround to preventing a particular kind memory limit error on relativistic simulations looks an awful lot like dark matter. Someone else shows that finite precision errors in quantum field theory simulations can lead to the asymmetry in the abundance of matter over anti-matter. All of these accumulating studies begin to build a picture of precisely how the universal simulation runs, and what compromises were forced on its design. Decades after the inciting observations, the paradigm shift is completed. Nobody wants to admit it, but it fits all the explanations together, and the Simulation Hypothesis Theory is the last explanation left standing. 781