fix(dog): spawn session and set BD_ACTOR for dog dispatch

Recovered from reflog - these commits were lost during a rebase/force-push.

Dogs are directories with state files but no sessions. When `gt dog dispatch`
assigned work and sent mail, nothing executed because no session existed.

Changes:
1. Spawn tmux session after dispatch (gt-<town>-deacon-<dogname>)
2. Set BD_ACTOR=deacon/dogs/<name> so dogs can find their mail
3. Add dog case to AgentEnv for proper identity

Session spawn is non-blocking - if it fails, mail was sent and human can
manually start the session.

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

internal/config/env_test.go

@@ -125,6 +125,23 @@ func TestAgentEnv_Boot(t *testing.T) {
 	assertNotSet(t, env, "BEADS_NO_DAEMON")
 }
 
+func TestAgentEnv_Dog(t *testing.T) {
+	t.Parallel()
+	env := AgentEnv(AgentEnvConfig{
+		Role:      "dog",
+		AgentName: "alpha",
+		TownRoot:  "/town",
+	})
+
+	assertEnv(t, env, "GT_ROLE", "dog")
+	assertEnv(t, env, "BD_ACTOR", "deacon/dogs/alpha")
+	assertEnv(t, env, "GIT_AUTHOR_NAME", "dog-alpha")
+	assertEnv(t, env, "GIT_AUTHOR_EMAIL", "dog-alpha@gastown.local")
+	assertEnv(t, env, "GT_ROOT", "/town")
+	assertNotSet(t, env, "GT_RIG")
+	assertNotSet(t, env, "BEADS_NO_DAEMON")
+}
+
 func TestAgentEnv_WithRuntimeConfigDir(t *testing.T) {
 	t.Parallel()
 	env := AgentEnv(AgentEnvConfig{