$ cat e.py
raise Exception
$ python e.py
Traceback (most recent call last):File "e.py", line 1, in <module>raise Exception
Exception
$ echo $?
1

I would like to change this exit code from 1 to 3 while still dumping the full stack trace. What's the best way to do this?

Take a look at the traceback module. You could do the following:

import sys, tracebacktry:raise Exception()
except:traceback.print_exc()sys.exit(3)

This will write traceback to standard error and exit with code 3.