Stabilizes the Responses API proxy header test by splitting the coverage
at the right boundary:
- Core integration test now verifies parent/subagent identity headers
directly from captured `/responses` requests.
- Proxy dump unit test now verifies those identity headers are preserved
in dumped request JSON.
- Removes the flaky real proxy process + temp-file dump polling path
from the core test.
This makes Responses API proxy request/response dumping first-class by
adding an optional `--dump-dir` flag that emits paired JSON files with
shared sequence/timestamp prefixes, captures full request and response
headers and records parsed JSON bodies.