[Bugfix][Structured Output] Preserve reasoning adapters for shared parser engines

[sfeng33]

Purpose

Fix structured-output reasoning configuration being silently disabled when the reasoning and tool parsers use the same parser engine.

ParserManager.get_parser() returned the raw shared ParserEngine, whose reasoning_parser attribute is None. The Chat Completions and Responses serving paths interpreted this as reasoning being disabled and skipped forwarding
request-specific parser kwargs such as enable_thinking=False.

The structured-output backend could then use the parser's default thinking mode and wait for a reasoning boundary that the model was instructed not to emit, leaving schema generation unconstrained or running until max_tokens.

Test Plan

Case	STRICT	main (before)	PR (after)
required, non-stream	1	no tool_calls	✅ get_weather {"city": "San Francisco"}
required, stream	1	no tool_calls	✅ get_weather {"city": "San Francisco"}
named, non-stream	1	no tool_calls	✅ get_weather {"city": "San Francisco"}
named, stream	1	no tool_calls	✅ get_weather {"city": "San Francisco"}
required, non-stream	0	no tool_calls	✅ get_weather {"city": "San Francisco"}
named, non-stream	0	no tool_calls	✅ get_weather {"city": "San Francisco"}
named, stream	0	no tool_calls	✅ get_weather { "city": "San Francisco" }
auto, non-stream	0,1	content: "" (24 tokens dropped)	✅ content: "<tool_call>\n{"name": "get_weather", ...}\n</tool_call>"
auto, non-stream, no-think	0,1	content: ""	✅ same content restored
auto, non-stream, include_reasoning:false	0	content: ""	✅ same content restored

[claude]

Claude Code Review

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[sfeng33]

/ci run

[github-actions]

✅ Triggered Buildkite CI #85559 for commit 553201d58097.

[yewentao256]

LGTM, thanks for the work!