ci(api): run handler integration tests

Keep localhost-dependent handler coverage active in the Docker integration job.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
Copilot-Session: 5e77d279-9f4a-46bc-97b1-db6975c76faf

Author: AchoArnold

.github/workflows/api.yml

@@ -74,6 +74,12 @@ jobs:
           docker compose wait seed || true
           sleep 2
 
+      - name: Run Handler Integration Tests
+        working-directory: ./api
+        env:
+          USER_API_KEY: test-user-api-key
+        run: go test -tags integration -v -timeout 60s ./pkg/handlers
+
       - name: Run Integration Tests
         working-directory: ./tests
         run: go test -v -timeout 300s ./...