refactor: drop persist_session plumbing + fix broken btw mid-turn bypass (#16075)

Follow-up to PR #16053 (/btw as /background alias). Cleans up the
plumbing added exclusively for the old ephemeral /btw handler and
repairs a broken btw bypass that landed between my refactor and this
follow-up.

run_agent.py:
- Remove persist_session kwarg, instance attr, and _persist_session
  short-circuit. Only /btw ever passed persist_session=False; with
  /btw gone the default (always persist) is the only behavior anyone
  ever wanted.

gateway/run.py:
- Remove the unreachable 'if _cmd_def_inner.name == "btw"' block
  (PR #16059). Canonical name for a /btw message is 'background' after
  alias resolution — the comparison could never be true, and it called
  _handle_btw_command which no longer exists. The /background branch
  above it already dispatches /btw correctly.

tests/gateway/test_running_agent_session_toggles.py:
- Fix test_btw_dispatches_mid_run to mock _handle_background_command
  (the real dispatch target for /btw) instead of the deleted
  _handle_btw_command.

run_agent.py

@@ -892,7 +892,6 @@ class AIAgent:
         checkpoints_enabled: bool = False,
         checkpoint_max_snapshots: int = 50,
         pass_session_id: bool = False,
-        persist_session: bool = True,
     ):
         """
         Initialize the AI Agent.
@@ -964,7 +963,6 @@ class AIAgent:
         self.background_review_callback = None  # Optional sync callback for gateway delivery
         self.skip_context_files = skip_context_files
         self.pass_session_id = pass_session_id
-        self.persist_session = persist_session
         self._credential_pool = credential_pool
         self.log_prefix_chars = log_prefix_chars
         self.log_prefix = f"{log_prefix} " if log_prefix else ""
@@ -3353,10 +3351,7 @@ class AIAgent:
         """Save session state to both JSON log and SQLite on any exit path.
 
         Ensures conversations are never lost, even on errors or early returns.
-        Skipped when ``persist_session=False`` (ephemeral helper flows).
         """
-        if not self.persist_session:
-            return
         self._apply_persist_user_message_override(messages)
         self._session_messages = messages
         self._save_session_log(messages)