If the breaker is suspect, it's probably easier and cheaper to replace it than to test it. They cost less than $20.
Most electricians would not be suspicious of the breaker unless there were heat damage at more than one point in the shore power connection. Had an actual overload, rather than plug and socket problems, caused the damage, it would be present equally throughout the circuit, and burned or melted insulation would be present in the vicinity of the breaker as well as in the vicinity of the plug. If this is the case, all the damaged wiring should be replaced, in addition to the breaker.
Breakers that don't trip are rare (some industry sources estimate an 0.01% failure rate), aside from the notorious asymmetrical trip defect in older Federal-Pacific breakers.