[FEATURE] Claude apps gateway: support 1-hour prompt cache TTL

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• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Gateway sessions are limited to the 5-minute prompt cache TTL. The CLI omits the extended-cache-ttl beta on gateway sessions, so promptCacheTtl, CLAUDE_CODE_PROMPT_CACHE_TTL and ENABLE_PROMPT_CACHING_1H have no effect (availability table, prompt caching).

As a result, any idle gap over 5 minutes rewrites the full context at the cache-write rate. The same session costs more through the gateway than on direct Bedrock with ENABLE_PROMPT_CACHING_1H=1.

The documented reason is that not every upstream supports 1h. The gateway knows its upstreams, though, and ours is Bedrock, which supports 1h for every current Claude model.

Proposed Solution

Support the 1-hour TTL on gateway sessions when the configured upstream supports it.

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

API and model interactions

Use Case Example

_No response_

Additional Context

Claude Code 2.1.285 (gateway and clients), provider: bedrock, us-west-2. Related: #74075, #89488.