Proofs exist only in mathematics where a theory starts with axioms and theorems are proven with a quod erat demonstrandum .
Physics uses mathematics for modeling, imposing postulates that relate mathematical quantities and functions to measurements and observations. This has been an ongoing progress ever since Newton and Maxwell.
Physicists work on theories that are based on the big bang theory.
They are based on the Big Bang Model within the theory of General Relativity. It has already changed as a model from the time I took a cosmology course in the 1970s .
Quantum mechanics is a theory, Classical mechanics is a theory, Thermodynamics is a theory, Classical electrodynamics is a theory.
There exist continuities in the theories of Physics. Thermodynamics emerges from statistical mechanics. Classical electromagnetic waves emerge from a huge number of photons. General Relativity for low masses becomes Newtonian mechanics etc.
If the observations and experiments are correctly described by a model based on a theory, and a new datum/observation falsifies not only the model but a basic premise/postulate of the theory on which the model is based, as happened with classical theories and the need for quantum mechanics, (black body radiation, the photoelectric effect) the new theory will have a region of validity at the boundaries of which it should agree with the old theory , plus explain the discrepancy and predict new phenomena.
However, if the big bang was disproved sometime in the future, all the related theories will too. Isn't that possible to happen to one of the theories that we currently take for granted?
No, related models will have to change, but the theory of General Relativity will not need to change if the BB is falsified, i.e. there is an observation disagreeing with the predictions of the model. A new model will have to be formulated that will incorporate all the data and observations plus explain the discrepancy that invalidated the BB model. The theory of GR will have to change if observations/data are found in some region of parameters and variables where the postulates are falsified. The prime example is the implicit postulate of continuous values in energy in classical thermodynamics with the quantization needed to explain black body radiation that eventually forced quantum theory to appear. The Bohr atom has as a postulate the quantization, except that it is a model not a theory. In the final theory quantization appears as a consequence of the postulates.