Can the conservation of the 4-momentum be proved, or is it a postulate of special relativity?
If by "proved" you mean "experimentally verified", yes, tons of particle physics experiments have verified conservation of 4-momentum to high precision. (Not to mention tons of other experiments plus common experience, but most data other than particle physics is limited to the non-relativistic regime.)
As far as being a "postulate" of SR, yes, it basically is; if you want to be precise, it's a necessary consequence of the postulates of SR, since those postulates imply flat spacetime, and conservation of 4-momentum is a consequence of flat spacetime. (More specifically, conservation of 4-momentum is a consequence of time and space translation invariance, i.e., it holds in any spacetime with those properties.)
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