Why not use physical observation as proof? At least it can be observed by all who wish to see for themselves.
We have observed the red-shifting of light with galaxies as their distance increases from us. It has been measured and widely agreed upon that space-time is still expanding at an ever increasing rate. I would choose this observation as the basis for the existence of time. Please note that this observation shows an inseparable relationship with space. Manipulating one also influences the other.
And then of course there is one actual scientific experiment about space-time. It was called the Casimir effect named after the guy that discovered it. It was observed that two metal plates when brought extremely close to each other will suddenly attract very rapidly. It was interpreted to mean that space-time has pressure and can cause matter to move. The space-time between the plates is shielded by the plates from the space-time outside the plates thus reducing the available push effect. An imbalance in push force results in the apparent plate attraction.
Now the next step in understanding would rely on the above statements as being factually true. If the above is basic knowledge, we should be able to use that knowledge as building blocks for something more advanced. Something more advanced that comes from this basic knowledge would tend to validate the basic knowledge.
I posed myself a question as the next step. The question I asked myself is: Do I really need two metal plates to create the push effect with space-time?