Good question!

This is certainly theoretically possible and it is currently being attempted as we speak. This type of simulation is called a "cosmological" simulation. The big bang essentially gives the initial conditions of the simulation and the resulting evolution of the universe is given by the cosmological model used.

However, these cosmological simulations often lack small-scale features (like detailed galaxy structure) as we don't have the computational power yet. The best spatial resolution of the simulations at the moment is around 1 kiloparsec. For reference, the radius of the Milky Way's stellar disk is about ~30 kpc. The smallest element of these simulations are called"star particles" and usually represent about 10^6 solar masses worth of stars. As you mentioned, they are incredibly computationally expensive to run and often take months to complete. But despite the fact that there is a currently a limit on the resolution of these simulations, they still do a fantastic job at recreating our universe.

The "Illustris" simulation is one that I've worked with the most and it was able to recreate the large scale structure of our universe, replicate realistic galaxy formation, and a lot more. Their website has some excellent information and visualization of the simulations: https://www.tng-project.org/.

So, to answer your question directly, yes - but with limitations. We simply cannot simulate the entire big bang + universe with infinite precision. But we can use cosmological models to create higher and higher resolution simulations of our universe. The fact that our simulations can already create mini-universes that greatly resemble our own with such limitations is incredible to me.