I want to redefine the bytecode of the StackOverflowError constructor so I have a "hook" for when a stack overflow occurs. All I want to do is insert a single method call to a static method of my choosing at the start of the constructor. Is it possible to do this?
You should be able to do it using one of two ways (unless something changed in the last 1-2 years, in which case I'd love some links to changelogs/docs):
Mentioned in a comment, not very feasible I guess, modify the classes you are interested in, put them in a jar and then use the -bootclasspath option to load them instead of the default ones. As was mentioned before this can have some legal issues (and is a pain to do in general).
You should be able to (or at least you used to be able to) instrument almost all core classes (iirc Class was the only exception I've seen). One of many problems you might have is the fact that many of core classes are being initialized before the agents you provide (or well their premain methods to be exact) are consulted. To overcome this you will have to add Can-Retransform-Classes property to your agent jar and then re-transform the classes you are interested in. Be aware that re-transformation is a bit less powerful and doesn't give you all the options you'd have normally with instrumentation, you can read more about it in the doc.
I am assuming you know how to do instrumentation?