browser example: add database usage to example [INT-355]

Author: perf2711

examples/sdk/browser/package-lock.json

@@ -14,6 +14,9 @@ const client = BacktraceClient.builder({
             prop2: 123,
         },
     },
+    database: {
+        enable: true,
+    },
 })
     .useModule(
         new BacktraceSessionReplayModule({