[ENHANCEMENT] OpenAI-Compatible provider: support Responses API so /responses-only models work

[kostia-ivashchenko]

Problem (one or two sentences)

Concretely, GPT-5.6-class models from several gateways now return:

{"error":{"code":"unsupported_api_for_model","message":"model \"gpt-5.6-sol\" is not accessible via the /chat/completions endpoint"}}

The same model, sent to /v1/responses, returns a normal 200 completion.

Context (who is affected and when)

Anyone routing the OpenAI-Compatible provider at a gateway that exposes
Responses-only models. This is increasingly common:

• GitHub Copilot OAuth (via a local OpenAI-compatible proxy) serves
  gpt-5.6-terra, gpt-5.6-sol, gpt-5.6-luna, gpt-5.5, gpt-5.4,
  gpt-5.3-codex only through /v1/responses.
• Codex serves its GPT coding models the same way.

These models are visible in the model list and selectable in Zoo Code, but
every request errors out — which is misleading, because the model appears
supported until it's actually used.

Desired behavior (conceptual, not technical)

The OpenAI-Compatible provider should be able to reach /v1/responses for
models that are served only on that endpoint, so those models complete
normally instead of erroring out. This is already how the Opencode Go provider
handles GPT-5.6 models (PR #1443); extending the same capability to the generic
OpenAI-Compatible provider would let Responses-only GPT models be used in
Zoo Code.

Constraints / preferences (optional)

• A per-model or per-profile opt-in toggle ("use Responses API") is probably
  the safest shape, since not all OpenAI-compatible gateways implement
  /v1/responses.
• Chat-compatible models (e.g. codex-auto-review) should stay on
  /chat/completions.
• The OpenAI-Compatible provider is in active development — see PR feat(api): abort signal support for openai-native and openai-compatible (completePrompt + createMessage) #1291
  (abort signal support for openai-native and openai-compatible), which touches
  the same code path.

Request checklist

• I've searched existing Issues and Discussions for duplicates
• This describes a specific problem with clear context and impact

Zoo Code Task Links (optional)

No response

Acceptance criteria (optional)

No response

Proposed approach (optional)

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Trade-offs / risks (optional)

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