Endless Rename-Item errors during Flutter Upgrade (Android Studio)

For the last few Flutter upgrades, I was unable to update from inside Android Studio.

## Steps to Reproduce

Flutter Upgrade from inside Android Studio gives an endless loop of the following errors. Leaving AS and upgrading from the Flutter console works flawlessly.

```
P:\Flutter\bin\flutter.bat --no-color upgrade
Upgrading Flutter to 3.24.3 from 3.24.2 in P:\Flutter...
Checking Dart SDK version...
Downloading Dart SDK from Flutter engine 36335019a8eab588c3c2ea783c618d90505be233...
Rename-Item : The process cannot access the file because it is being used by another process.
At P:\Flutter\bin\internal\update_dart_sdk.ps1:74 char:9
+ Rename-Item $dartSdkPath "$oldDartSdkPrefix$oldDartSdkSuffix"
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : WriteError: (P:\Flutter\bin\cache\dart-sdk:String) [Rename-Item], IOException
+ FullyQualifiedErrorId : RenameItemIOError,Microsoft.PowerShell.Commands.RenameItemCommand

Error: Unable to update Dart SDK. Retrying...
ERROR: Input redirection is not supported, exiting the process immediately.
Checking Dart SDK version...
Downloading Dart SDK from Flutter engine 36335019a8eab588c3c2ea783c618d90505be233...
Rename-Item : The process cannot access the file because it is being used by another process.
At P:\Flutter\bin\internal\update_dart_sdk.ps1:74 char:9
+ Rename-Item $dartSdkPath "$oldDartSdkPrefix$oldDartSdkSuffix"
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : WriteError: (P:\Flutter\bin\cache\dart-sdk:String) [Rename-Item], IOException
+ FullyQualifiedErrorId : RenameItemIOError,Microsoft.PowerShell.Commands.RenameItemCommand

Error: Unable to update Dart SDK. Retrying...
ERROR: Input redirection is not supported, exiting the process immediately.
Checking Dart SDK version...
```

## Version info

```
P:\Flutter\bin\flutter.bat doctor --verbose
[√] Flutter (Channel stable, 3.24.3, on Microsoft Windows [Version 10.0.22631.4169], locale en-US)
• Flutter version 3.24.3 on channel stable at P:\Flutter
• Upstream repository https://github.com/flutter/flutter.git
• Framework revision 2663184aa7 (2 weeks ago), 2024-09-11 16:27:48 -0500
• Engine revision 36335019a8
• Dart version 3.5.3
• DevTools version 2.37.3

[√] Windows Version (Installed version of Windows is version 10 or higher)

[√] Android toolchain - develop for Android devices (Android SDK version 34.0.0)
• Android SDK at P:/Android_SDK
• Platform android-34, build-tools 34.0.0
• ANDROID_HOME = P:/Android_SDK
• ANDROID_SDK_ROOT = p:\Android_SDK
• Java binary at: C:\Program Files\Android\Android Studio\jbr\bin\java
• Java version OpenJDK Runtime Environment (build 17.0.11+0--11852314)
• All Android licenses accepted.

[X] Chrome - develop for the web (Cannot find Chrome executable at .\Google\Chrome\Application\chrome.exe)
! Cannot find Chrome. Try setting CHROME_EXECUTABLE to a Chrome executable.

[√] Visual Studio - develop Windows apps (Visual Studio Community 2022 17.11.4)
• Visual Studio at C:\Program Files\Microsoft Visual Studio\2022\Community
• Visual Studio Community 2022 version 17.11.35312.102
• Windows 10 SDK version 10.0.22621.0

[√] Android Studio (version 2024.1)
• Android Studio at C:\Program Files\Android\Android Studio
• Flutter plugin can be installed from:
https://plugins.jetbrains.com/plugin/9212-flutter
• Dart plugin can be installed from:
https://plugins.jetbrains.com/plugin/6351-dart
• Java version OpenJDK Runtime Environment (build 17.0.11+0--11852314)

[√] Connected device (2 available)
• Windows (desktop) • windows • windows-x64 • Microsoft Windows [Version 10.0.22631.4169]
• Edge (web) • edge • web-javascript • Microsoft Edge 129.0.2792.52

[√] Network resources
• All expected network resources are available.

! Doctor found issues in 1 category.
Process finished with exit code 0
```