The closest we can get to faking time travel is by manipulating the speed of a subset of reactions (like our body's aging) by going the speed of light or freezing ourselves, etc. But it's still fake, "time" continues, you just managed to manipulate specific reactions in a peculiar way.
Also, can you explain what you think "physical reactions" are? What causes them? Why would they happen at different speeds if time doesn't exist?
Can you explain which of the two statements you disagree with and provide an explanation:
1. The speed of light is constant in all frames of reference.
2. The laws of Physics apply in all inertial frames of reference.
If you disagree with neither of these statements, then the entirety of special relativity must follow from your logic. Which means you have to explain why these physical effects happen without being able to explain it through time dilation and other effects.