[BUG] SSL certificate verification failed when routing via custom inference gateway / corporate proxy

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

When running Claude Code in environments that route traffic through a custom corporate API gateway or SSL inspection proxy, requests fail due to SSL certificate verification errors.

Specifically, when Claude Code is invoked via Claude Desktop / Claude Code integration (`isClaudeCodeForDesktopEnabled = true`) or directly against a custom inference gateway (`inferenceGatewayBaseUrl`), network calls to the model endpoint fail with an SSL verification error.

Tested versions with NO SSL error:
- 1.32885.1 (commit a757f5)
- 1.37937.3 (commit 28dcf5)

Tested versions exhibiting the SSL error:
- 1.40609.1 (commit 69aac0)
- 1.46388.4 (commit 50e62f)

System / Environment:
- Platform: win32 (x64)
- OS Version: Windows 10.0.26100
- CPU: 13th Gen Intel(R) Core(TM) i7-1360P (16 cores)
- Memory: 16 GB total, 2 GB available

Observation:
The regression was introduced between 1.37937.3 and 1.40609.1. Earlier builds respect system/gateway certificates or existing TLS configurations, whereas 1.40609.1 and all subsequent releases consistently throw SSL verification failures in the exact same enterprise environment.

### What Should Happen?

Claude Code should complete API requests through configured custom inference proxies/gateways without failing SSL verification, or properly inherit trusted root CAs from standard environment configuration (NODE_EXTRA_CA_CERTS, SSL_CERT_FILE, system trust store).

### Error Messages/Logs

```shell
API Error: Unable to connect to API: SSL certificate verification failed. Check your proxy or corporate SSL certificates
```

### Steps to Reproduce

1. Configure Claude Code in an environment that routes model inference via an enterprise proxy or API gateway (inferenceGatewayBaseUrl = "https://aigateway.example.com").
2. Execute any prompt or command with Claude Code (`claude "hello"` or via the Claude Code integration).
3. Observe the immediate failure: API Error: Unable to connect to API: SSL certificate verification failed.
4. Downgrade/switch back to version 1.37937.3 under identical network settings: requests succeed normally.
5. Upgrade to 1.40609.1 or 1.46388.4: SSL verification fails again.

### Claude Model

Sonnet (default)

### Is this a regression?

Yes, this worked in a previous version

### Last Working Version

1.37937.3

### Claude Code Version

Claude 1.52386.6 (612f3f) 2026-09-13T00:45:40.000Z

### Platform

Anthropic API

### Operating System

Windows

### Terminal/Shell

Windows Terminal

### Additional Information

Workarounds tested:

- What worked (insecure bypass only):
Setting NODE_TLS_REJECT_UNAUTHORIZED=0 bypasses the failure, confirming the issue is strictly root CA / SSL certificate validation in the client runtime.
- What did NOT work:
NODE_OPTIONS, NODE_USE_SYSTEM_CA, NODE_EXTRA_CA_CERTS (after explicitly exporting and pointing to the full corporate root CA bundle), and SSL_CERT_FILE. The runtime appears to ignore or fail to pass these CA options to the underlying HTTP client initiating the inference requests.

Relevant Configuration Details:
```
[deployment-mode] Effective mode: 3P
[deployment-mode] Persisted mode choice: 3p
[deployment-mode] inferenceProvider set: yes
[deployment-mode] Bootstrap configured: no
[managed-config] Source: local
[managed-config] inferenceGatewayBaseUrl = "https://aigateway.example.com"
[managed-config] isClaudeCodeForDesktopEnabled = true
[managed-config] isLocalDevMcpEnabled = true
[managed-config] autoModeEnabled = true
[managed-config] toolSearchEnabled = true
```