Hi!
I believe that found the problem, if the Live Coding configuration of integration is enable in your Unreal Engine 5, can be this that are creating the conflict of Mutex 
Try disable the Live Coding or press Ctrl+Alt+F11 in the editor(VS2022) or game, and try compile again.
This information are in a log file from Unreal Engine 5 in:
USERNAME/AppData/Local/UnrealBuildTool/Log.txt
See below the complete message:
Checking for live coding mutex: Global\LiveCoding_D++Epic Games+UE_5.1+Engine+Binaries+Win64+UnrealEditor.exe
Unable to build while Live Coding is active. Exit the editor and game, or press Ctrl+Alt+F11 if iterating on code in the editor or game
BuildException: Unable to build while Live Coding is active. Exit the editor and game, or press Ctrl+Alt+F11 if iterating on code in the editor or game
at UnrealBuildTool.HotReload.CheckForLiveCodingSessionActive(TargetDescriptor TargetDescriptor, TargetMakefile Makefile, BuildConfiguration BuildConfiguration, ILogger Logger) in D:\build++UE5\Sync\Engine\Saved\CsTools\Engine\Source\Programs\UnrealBuildTool\System\HotReload.cs:line 231
at UnrealBuildTool.BuildMode.Build(TargetMakefile[] Makefiles, List`1 TargetDescriptors, BuildConfiguration BuildConfiguration, BuildOptions Options, FileReference WriteOutdatedActionsFile, ILogger Logger) in D:\build++UE5\Sync\Engine\Saved\CsTools\Engine\Source\Programs\UnrealBuildTool\Modes\BuildMode.cs:line 526
at UnrealBuildTool.BuildMode.Execute(CommandLineArguments Arguments, ILogger Logger) in D:\build++UE5\Sync\Engine\Saved\CsTools\Engine\Source\Programs\UnrealBuildTool\Modes\BuildMode.cs:line 242
at UnrealBuildTool.UnrealBuildTool.Main(String[] ArgumentsArray) in D:\build++UE5\Sync\Engine\Saved\CsTools\Engine\Source\Programs\UnrealBuildTool\UnrealBuildTool.cs:line 648
WriteFileIfChanged() wrote 0 changed files of 0 requested writes.
have many more informations, but are personal information from my project and path from ssd 