We have an approximation

We have a fucking great deal more than an approximation. Newtonian physics were proven incorrect a century ago, but it's still 100% correct for everything going on on our planet.
Even a lander on Mars can be calculated correctly with Newtonian physics. It's not until the scale gets much larger that we need relativity.

we have no actual understanding of how things work at a fundamental level

That's moving the goal post. We have an extremely good understanding of how things work down to the quantum level, and that's pretty fundamental, or maybe you don't actually understand what quantum level means?
We can't measure below that because physics won't allow us. But it can also be said to be irrelevant, because no theory about it can be tested. It can be said to not actually be part of our universe. In much the same way the inside of a black hole is out of reach.
There are no groundbreaking things to be found there that are relevant to life on earth. If there were we would have hints about it, because they would permeate into our reality.

Hence there is no way to rule out anything definitively.

Technically true, but irrelevant regarding reality. It's the kind of thing religious nuts love to say, because you can't prove a negative.
There definitely is no god, and we can conclude that because there are no signs of a god, and if a god is crucial to our universe, we should see signs of it.
There are so many philosophical rules that confirm this, that it's a joke these questions are even debated.
The claim of cold fusion was a measurement error, there is no valid hypothesis and definitely not scientific theory that allows for cold fusion to exist.
And we know A LOT about atoms, enough to know how to make atom bombs for almost a century, and yet you use an argument from ignorance?!?!?!

mathematically.

You can describe a perfect circle very very easily mathematically, a circle with infinitely fine grained curvature. But such a circle cannot exist in reality, because quantum space prohibit it.

I don't give a shit about mathematical descriptions of things that aren't real, math is insanely efficient at describing reality, but it would be equally efficient at describing ANY reality.

Also there are no ways that math within the confines of the standard model can describe faster than light travel. Because in reality it's not just light, it is causality of everything that has a maximum speed of light. The idea of for instance tachyons moving backwards in time is only a hypothesis, not a scientific theory. There is nothing that shows that tachyons should exist. But many things that shows us they don't. Breaking down causality for one, and the same is the case with faster than light travel, it breaks causality, and makes the universe a Picasso universe without rules. And clearly we do not live in a Picasso universe, because there are indeed rules we can rely on for science to work.