fix(proxy): source takeover model fields from target provider on managed accounts

When a managed-account provider (GitHub Copilot or Codex OAuth) is taken
over by the local proxy, the *_MODEL_NAME entries written to Claude live
config used to be snapshotted from the live file on disk, which still
held the *previous* provider's values during a switch. The result was a
stale display name lingering after the switch.

Read the model snapshot from the target provider's effective settings
(common-config merged) when the provider uses managed-account auth;
preserve the existing behavior of reading from the live config for
normal providers. The three takeover entry points now resolve the
effective provider up front so common-config writes flow through.