feat(ci): Add integration test job; refactor crew startup to use beacon prompt

CI: Add integration test job that runs go test -tags=integration for
install, rig, and beads routing tests.

Crew lifecycle: Pass startup beacon as Claude's initial prompt instead
of nudging after startup. Removes timing-dependent sleep/nudge sequence.
Also removes redundant SetEnvironment calls (env vars already exported
in BuildCrewStartupCommand).

🤖 Generated with Claude Code

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

.github/workflows/ci.yml

@@ -71,3 +71,31 @@ jobs:
         with:
           version: latest
           args: --timeout=5m
+
+  integration:
+    name: Integration Tests
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v6
+
+      - name: Set up Go
+        uses: actions/setup-go@v5
+        with:
+          go-version: '1.24'
+
+      - name: Configure Git
+        run: |
+          git config --global user.name "CI Bot"
+          git config --global user.email "ci@gastown.test"
+
+      - name: Install beads (bd)
+        run: go install github.com/steveyegge/beads/cmd/bd@latest
+
+      - name: Build gt
+        run: go build -v -o gt ./cmd/gt
+
+      - name: Add to PATH
+        run: echo "$(go env GOPATH)/bin" >> $GITHUB_PATH
+
+      - name: Integration Tests
+        run: go test -tags=integration -timeout=5m -v ./internal/cmd/...