Flutter widget unit test - expect exception

Question

I have the following unit test, which intentionally throws an error.

Note: this is a minimal example and not representative of the actual problem I'm trying to solve.

import 'dart:async';

import 'package:flutter/material.dart';
import 'package:flutter_test/flutter_test.dart';

void main() {
  testWidgets('future builder ex error', (tester) async {
    await tester.pumpWidget(MaterialApp(
        home: Scaffold(
            body: FutureBuilder<void>(
                future: delayedError(),
                builder: (context, _) => const Text('hi')))));

    await tester.pumpAndSettle(const Duration(seconds: 3));
    expect(tester.takeException(), isInstanceOf<Exception>());
  });
}

Future<void> delayedError() async {
  late final Timer timer;
  timer = Timer.periodic(const Duration(seconds: 1), (t) {
    timer.cancel();
    throw Exception('Something bad happened');
  });
}

When I run this code I'm seeing the following output:

═╡ EXCEPTION CAUGHT BY FLUTTER TEST FRAMEWORK ╞════════════════════════════════════════════════════
The following _Exception was thrown running a test:
Exception: Something bad happened

When the exception was thrown, this was the stack:
#0      delayedError.<anonymous closure> (file:///home/bsutton/git/future_builder_ex/test/src/future_builder_ex2_test.dart:23:5)
#8      FakeTimer._fire (package:fake_async/fake_async.dart:323:16)
#9      FakeAsync._fireTimersWhile (package:fake_async/fake_async.dart:244:13)
#10     FakeAsync.elapse (package:fake_async/fake_async.dart:141:5)
#11     AutomatedTestWidgetsFlutterBinding.pump.<anonymous closure> (package:flutter_test/src/binding.dart:1047:28)
#14     TestAsyncUtils.guard (package:flutter_test/src/test_async_utils.dart:71:41)
#15     AutomatedTestWidgetsFlutterBinding.pump (package:flutter_test/src/binding.dart:1043:27)
#16     WidgetTester.pumpAndSettle.<anonymous closure> (package:flutter_test/src/widget_tester.dart:675:23)
#19     TestAsyncUtils.guard (package:flutter_test/src/test_async_utils.dart:71:41)
#20     WidgetTester.pumpAndSettle (package:flutter_test/src/widget_tester.dart:668:27)
#21     main.<anonymous closure> (file:///home/bsutton/git/future_builder_ex/test/src/future_builder_ex2_test.dart:14:18)
<asynchronous suspension>
<asynchronous suspension>
(elided 12 frames from dart:async and package:stack_trace)

The test description was:
  future builder ex error
════════════════════════════════════════════════════════════════════════════════════════════════════
Test failed. See exception logs above.
The test description was: future builder ex error

══╡ EXCEPTION CAUGHT BY FLUTTER TEST FRAMEWORK ╞════════════════════════════════════════════════════
The following TestFailure was thrown running a test (but after the test had completed):
Expected: <Instance of 'Exception'>
  Actual: <null>
   Which: is not an instance of 'Exception'

When the exception was thrown, this was the stack:
#0      fail (package:test_api/src/expect/expect.dart:134:31)
#1      _expect (package:test_api/src/expect/expect.dart:129:3)
#2      expect (package:test_api/src/expect/expect.dart:46:3)
#3      expect (package:flutter_test/src/widget_tester.dart:460:16)
#4      main.<anonymous closure> (file:///home/bsutton/git/future_builder_ex/test/src/future_builder_ex2_test.dart:15:5)
<asynchronous suspension>
<asynchronous suspension>
(elided one frame from package:stack_trace)
════════════════════════════════════════════════════════════════════════════════════════════════════
✖ future builder ex error

I think the core of my problem resolves around the second stack trace:

The following TestFailure was thrown running a test (but after the test had completed):
Expected: <Instance of 'Exception'>
  Actual: <null>
   Which: is not an instance of 'Exception'

I'm guessing that the exception is not being captured by the framework so test.takeException doesn't see an error.

How do I setup widget unit tests that are intended to check that exceptions are being correctly handled?