Claude provider needs a Shadow config path, matching Codex, for multi-account thread continuation

[hangtohi-tech]

Problem

Codex providers support "Shadow home path" (docs/providers/codex.md): two Codex
instances that share the same CODEX_HOME path but differ only by Shadow home path are treated as "compatible for continuation" — the locked model picker in
an existing thread shows both, so switching accounts mid-thread (e.g. when one
account hits its usage limit) just works.

The Claude provider has no equivalent. A Claude instance's settings only expose:
Environment variables, Binary path, CLAUDE_CONFIG_DIR path, Launch arguments —
there is no "Shadow config path" field. Two Claude instances pointed at different
CLAUDE_CONFIG_DIR values are always treated as separate workspaces, so an
existing thread shows "provider is unavailable in this thread. Start a new
thread to switch providers." even though both instances are Claude Pro/Max
accounts on the same underlying CLI.

This matches the technical detail from #4696 (vedmalex's comment): the thread
continuation key for Claude is claude:home:<resolved path> — a straight path
comparison, no shadow/alias concept, unlike Codex's adapter.

Why this matters

Common workflow: a user has two Claude Pro/Max subscriptions (personal + work,
or just two accounts for quota headroom) and wants to switch to the second
account mid-thread when the first hits its 5h/weekly limit, without losing
conversation context — exactly the use case Codex already supports via Shadow
home path.

Suggested fix

Port the same "shared home + shadow path" pattern from the Codex adapter to the
Claude adapter:

• Add a Shadow config path field to the Claude provider instance settings
  (parallel to Codex's Shadow home path)
• When set, resolve credentials/auth from the shadow directory while keeping the
  thread continuation key tied to the primary CLAUDE_CONFIG_DIR path, so two
  Claude instances sharing a primary path are considered continuation-compatible
  the same way Codex instances are.

Related: #1444, #2111 (general Claude multi-account requests, already
addressed for the "can I configure multiple accounts" question by #4017 —
this issue is specifically about mid-thread continuation, which is still open).