Wire dart2js through flutter tool, add compilation test (#27668)

diff --git a/dev/bots/test.dart b/dev/bots/test.dart
index 41dd11e..947d0d02 100644
--- a/dev/bots/test.dart
+++ b/dev/bots/test.dart
@@ -167,10 +167,22 @@
     await _flutterBuildApk(path);
     await _flutterBuildIpa(path);
   }
+  await _flutterBuildDart2js(path.join('dev', 'integration_tests', 'web'));
 
   print('${bold}DONE: All build tests successful.$reset');
 }
 
+Future<void> _flutterBuildDart2js(String relativePathToApplication) async {
+  print('Running Dart2JS build tests...');
+  await runCommand(flutter,
+    <String>['build', 'web', '-v'],
+    workingDirectory: path.join(flutterRoot, relativePathToApplication),
+    expectNonZeroExit: false,
+    timeout: _kShortTimeout,
+  );
+  print('Done.');
+}
+
 Future<void> _flutterBuildAot(String relativePathToApplication) async {
   print('Running AOT build tests...');
   await runCommand(flutter,