The answer from @Tamas Tárnok led me down the right path, so I will mark he answer as the solution.

It seems that if you use job.Status.Failed = true, the exception message will actually be displayed in a popup. Let me demonstrate:

public class ImportCommand : Command
{
    private readonly ICommandArgFactory _commandArgFactory;

    public ImportCommand(
        ICommandArgFactory commandArgFactory
    {
        if (commandArgFactory == null) throw new ArgumentNullException(nameof(commandArgFactory));
        _commandArgFactory = commandArgFactory;
    }

    public override void Execute(CommandContext context)
    {
        NameValueCollection parameters = _commandArgFactory.GetArgs(context);

        Sitecore.Context.ClientPage.Start(this, "Import", new ClientPipelineArgs(parameters));
    }

    private void Import(ClientPipelineArgs args)
    {
        if (args.IsPostBack)
        {
            if (args.Result == "no") return;

            ProgressBox.Execute("Import", "Import", DoImport, args);
        }
        else
        {
            SheerResponse.Confirm("Are you sure?");
            args.WaitForPostBack();
        }
    }

    private void DoImport(object[] parameters)
    {
        BaseJob job = Sitecore.Context.Job;

        if (job == null) return;

        try
        {
            ClientPipelineArgs args = parameters[0] as ClientPipelineArgs;

            job.Status.State = JobState.Running;
            ...code here...
            job.Status.AddMessage("Parsing some more...");
            job.Status.State = JobState.Finished;
        }
        catch (CustomException customEx)
        {
            Log.Error(customEx.Message, this);
            job.Status.LogError(customEx.Message);
            job.Status.Failed = true;
        }
        catch (Exception ex)
        {
            Log.Error(ex.Message, this);
            job.Status.LogError(ex.Message);
            job.Status.Failed = true;
        }
    }
}