Here is some modifications that you can add to DiscUtils to support FAT LFNs:

First, make these changes to the Fat\Directory.cs file, like this (you need to add the _lfns variable, the GetLfnChunk function, and modify the existing LoadEntries function to add the lines marked with //+++ below):

internal Dictionary<string, string> _lfns = new Dictionary<string, string>(StringComparer.OrdinalIgnoreCase);

private static string GetLfnChunk(byte[] buffer)
{
    // see http://home.teleport.com/~brainy/lfn.htm
    // NOTE: we assume ordinals are ok here.
    char[] chars = new char[13];
    chars[0] = (char)(256 * buffer[2] + buffer[1]);
    chars[1] = (char)(256 * buffer[4] + buffer[3]);
    chars[2] = (char)(256 * buffer[6] + buffer[5]);
    chars[3] = (char)(256 * buffer[8] + buffer[7]);
    chars[4] = (char)(256 * buffer[10] + buffer[9]);

    chars[5] = (char)(256 * buffer[15] + buffer[14]);
    chars[6] = (char)(256 * buffer[17] + buffer[16]);
    chars[7] = (char)(256 * buffer[19] + buffer[18]);
    chars[8] = (char)(256 * buffer[21] + buffer[20]);
    chars[9] = (char)(256 * buffer[23] + buffer[22]);
    chars[10] = (char)(256 * buffer[25] + buffer[24]);

    chars[11] = (char)(256 * buffer[29] + buffer[28]);
    chars[12] = (char)(256 * buffer[31] + buffer[30]);
    string chunk = new string(chars);
    int zero = chunk.IndexOf('\0');
    return zero >= 0 ? chunk.Substring(0, zero) : chunk;
}

private void LoadEntries()
{
    _entries = new Dictionary<long, DirectoryEntry>();
    _freeEntries = new List<long>();

    _selfEntryLocation = -1;
    _parentEntryLocation = -1;

    string lfn = null;  //+++
    while (_dirStream.Position < _dirStream.Length)
    {
        long streamPos = _dirStream.Position;
        DirectoryEntry entry = new DirectoryEntry(_fileSystem.FatOptions, _dirStream);

        if (entry.Attributes == (FatAttributes.ReadOnly | FatAttributes.Hidden | FatAttributes.System | FatAttributes.VolumeId))
        {
            // Long File Name entry
            _dirStream.Position = streamPos;  //+++
            lfn = GetLfnChunk(Utilities.ReadFully(_dirStream, 32)) + lfn;  //+++
        }
        else if (entry.Name.IsDeleted())
        {
            // E5 = Free Entry
            _freeEntries.Add(streamPos);
            lfn = null; //+++
        }
        else if (entry.Name == FileName.SelfEntryName)
        {
            _selfEntry = entry;
            _selfEntryLocation = streamPos;
            lfn = null; //+++
        }
        else if (entry.Name == FileName.ParentEntryName)
        {
            _parentEntry = entry;
            _parentEntryLocation = streamPos;
            lfn = null; //+++
        }
        else if (entry.Name == FileName.Null)
        {
            // Free Entry, no more entries available
            _endOfEntries = streamPos;
            lfn = null; //+++
            break;
        }
        else
        {
            if (lfn != null) //+++
            { //+++
                _lfns.Add(entry.Name.GetDisplayName(_fileSystem.FatOptions.FileNameEncoding), lfn); //+++
            } //+++
            _entries.Add(streamPos, entry);
            lfn = null; //+++
        }
    }
}

Second, add these two public functions to the Fat\FatFileSystem.cs file. They will be the new APIs to query on LFNs:

/// <summary>
/// Gets the long name of a given file.
/// </summary>
/// <param name="shortFullPath">The short full path to the file. Input path segments must be short names.</param>
/// <returns>The corresponding long file name.</returns>
public string GetLongFileName(string shortFullPath)
{
    if (shortFullPath == null)
        throw new ArgumentNullException("shortFullPath");

    string dirPath = Path.GetDirectoryName(shortFullPath);
    string fileName = Path.GetFileName(shortFullPath);
    Directory dir = GetDirectory(dirPath);
    if (dir == null)
        return fileName;

    string lfn;
    if (dir._lfns.TryGetValue(Path.GetFileName(shortFullPath), out lfn))
        return lfn;

    return fileName;
}

/// <summary>
/// Gets the long path to a given file.
/// </summary>
/// <param name="shortFullPath">The short full path to the file. Input path segments must be short names.</param>
/// <returns>The corresponding long file path to the file or null if not found.</returns>
public string GetLongFilePath(string shortFullPath)
{
    if (shortFullPath == null)
        throw new ArgumentNullException("shortFullPath");

    string path = null;
    string current = null;
    foreach (string segment in shortFullPath.Split(Path.DirectorySeparatorChar))
    {
        if (current == null)
        {
            current = segment;
            path = GetLongFileName(current);
        }
        else
        {
            current = Path.Combine(current, segment);
            path = Path.Combine(path, GetLongFileName(current));
        }
    }
    return path;
}

And that's it. Now, youll be able to dump a whole FAT disk recursively like this, for example:

static void Main(string[] args)
{
    using (FileStream fs = File.Open("fat.ima", FileMode.Open))
    {
        using (FatFileSystem floppy = new FatFileSystem(fs))
        {
            Dump(floppy.Root);
        }
    }
}

static void Dump(DiscDirectoryInfo di)
{
    foreach (DiscDirectoryInfo subdi in di.GetDirectories())
    {
        Dump(subdi);
    }
    foreach (DiscFileInfo fi in di.GetFiles())
    {
        Console.WriteLine(fi.FullName);
        // get LFN name
        Console.WriteLine(" " + ((FatFileSystem)di.FileSystem).GetLongFileName(fi.FullName));


        // get LFN-ed full path
        Console.WriteLine(" " + ((FatFileSystem)di.FileSystem).GetLongFilePath(fi.FullName));
    }
}

Use at your own risks! :)