The Big Bang and theism are in no way mutually exclusive

I agree

As for predictions, there are various prophecies and suchlike in the old testament that were fulfilled in the new testament - though you will probably call doubt onto the reliability of these sources.

Those are predictions about the past, not about the future. If you accept this as evidence, have you studied a large amount of other religions to see if they make similar claims? And if you had lived before the events of the new testament, would you have been able to distinguish the (supposedly true) "old testament" predictions from the (false) predictions made by thousands of other prophets of thousands other religions/sects?

My understanding of quantum physics is not as good as I wish it were - what is different?

First, radio-active decay is uncaused (or so it appears, to the extent that quantum mechanics is correct). But that is not the point I was trying to make.

In (special) relativity, "causality" basically refers to "all observers agree about light cones" or "if information flows from space-time point a to space-time point b, all observers will agree that the information went from a to b instead of vice versa", or simply "No one observes information travelling to the past". No notion of "all effects have a cause" or anything like that.

In mechanics, say the description of a billiard table, (naive) causality seems obvious: why did this ball change its direction? Because it was hit by another ball! all the way back to a player hitting the first ball. But, there is the principle of least action (the related principle of least time is better known but describes light only). The principle of least action says that the trajectory of all those billiard balls is a stationary point of some functional (the action). In this view, there is nothing special about ball collisions. We ask "why did the ball change its direction (when it was hit by another ball or by the wall)" but we might as well ask "why didn't the ball change its direction (when it wasn't hit by another ball or by the wall)" and the first question is no more natural than the second. And the answer to both questions is the same: "because that path is a stationary point of the action functional".

The naive notion of causality requires a known "natural" or "unperturbed" state. Only if we already know Newton's first law, then we can be surprised at the ball's change of direction. How would we apply this to the beginning of the universe? If you don't already know the natural unperturbed behaviour of the universe at/very near the big bang, how can you justify being surprised that it behaved the way it behaved? So surprised, in fact, that you have to imagine one or more high-complexity extra-universal entity(s) kick-starting the process?

So using scientific notions of causality near the big bang will get you nowhere - they don't say anything about causes, just about the order of certain events.