I managed to get it sort of working with Godot Mono! Just follow the general C# integration guide and then hook your client up to Godot's global exception handler like this:
GD.UnhandledException += (sender, args) => {
backtraceClient.Send(new BacktraceReport(args.Exception));
};It probably isn't as helpful as a full integration since you'll lose the stack trace whenever execution goes into Godot's native C++ code, but you'll still be able to log any exceptions that happen in your C# game logic.