Is there a pre-existing universe

How do we know the Big Bang didn't happen in an existing universe?

The current model of cosmology, which begins with the Big Bang, states that all spatial and temporal dimensions (length, width, height and time) as well as the four basic forces come from a single point. With the time that began with the Big Bang, there was no “before” just as there was no up and down, no gravity, or no electromagnetism. There was no universe in existence when the Big Bang occurred. It happened and then the universe existed (causality).

To say it started in another universe would require a different hypothesis than the current cosmological model.

There are many challenges in formulating a hypothesis that might explain what you are asking. Some of these challenges are to consider the laws of conservation of mass and energy. For example - if there was a "sparsely populated" pre-existing universe, then where did the mass and energy for our universe come from? Was it always there? If so, it must be some kind of black hole. So much should be taken for granted - we can have black holes that only exist in our universe with the smallest fraction of the total mass. If all of the mass were concentrated in one place, it would also exist within its own Schwarzschild radius. As a side note, a common exercise in college astronomy class is calculating the Schwarzschild radius of the observable universe. Surprisingly, our observable universe can be calculated as existing within an event horizon, but these calculations require many assumptions about the total mass and maximum extent of the observable universe (which usually limits the diameter to less than the Schwarzschild radius before starting the experiment).

Another difficulty could be to explain a black hole that suddenly goes in the opposite direction and expels all matter / energy, spatial dimensions, fundamental forces and time into an existing universe. How then does this explanation rule out any black holes believed to exist in the current epoch of our universe?

In short, we do not know that the universe we see today did not already begin in a universe that was in an existing state. However, to the best of my knowledge, no attempts have been made to explain how a sparsely populated universe could suddenly grow a densely populated mini-universe within it. All the attempts there may have been insufficient audiences to include me.

Also, remember that a hypothesis must be testable in order for it to become a theory. There is no way to test events that occurred before the Big Bang. Hence, there is no way to test the sparsely populated universe you asked about in your question.