¿Busca disponibilidad con MySQL (y PHP)?

Aquí hay una solución que parece funcionar:

SELECT t.*, DATEDIFF(t.LatestAvailable, t.EarliestAvailable) AS LengthAvailable
FROM 
   (SELECT u.*,
      COALESCE(b1.End, @StartOfWindow) AS EarliestAvailable,
      COALESCE(b2.Start, @EndOfWindow) AS LatestAvailable
    FROM LettingUnits u
    LEFT OUTER JOIN LettingUnitBookings b1
      ON (u.ID = b1.F_LU_ID AND b1.End BETWEEN @StartOfWindow AND @EndOfWindow)
    LEFT OUTER JOIN LettingUnitBookings b2
      ON (u.ID = b2.F_LU_ID AND b2.Start BETWEEN @StartOfWindow AND @EndOfWindow
          AND b2.Start >= b1.End) -- edit: new term
    ) AS t
LEFT OUTER JOIN LettingUnitBookings x
  ON (t.ID = x.F_LU_ID AND x.Start < t.LatestAvailable AND x.End > t.EarliestAvailable)
WHERE x.ID IS NULL AND DATEDIFF(t.LatestAvailable, t.EarliestAvailable) >= @WindowSize;

La salida es:

+-----+-------------+-------------------+-----------------+-----------------+
| ID  | Name        | EarliestAvailable | LatestAvailable | LengthAvailable |
+-----+-------------+-------------------+-----------------+-----------------+
| 123 | Foo Cottage | 2009-01-05        | 2009-01-10      |               5 |
| 123 | Foo Cottage | 2009-01-20        | 2009-01-25      |               5 |
| 456 | Bar Cottage | 2009-01-20        | 2009-01-31      |              11 |
+-----+-------------+-------------------+-----------------+-----------------+

Analizando esto con EXPLAIN muestra que emplea índices bastante bien:

+----+-------------+------------+--------+---------------+---------+---------+-------+------+-------------------------+
| id | select_type | table      | type   | possible_keys | key     | key_len | ref   | rows | Extra                   |
+----+-------------+------------+--------+---------------+---------+---------+-------+------+-------------------------+
|  1 | PRIMARY     | <derived2> | ALL    | NULL          | NULL    | NULL    | NULL  |    9 | Using where             |
|  1 | PRIMARY     | x          | ref    | F_LU_ID       | F_LU_ID | 8       | t.ID  |    2 | Using where; Not exists |
|  2 | DERIVED     | u          | system | NULL          | NULL    | NULL    | NULL  |    1 |                         |
|  2 | DERIVED     | b1         | ref    | F_LU_ID       | F_LU_ID | 8       | const |    0 |                         |
|  2 | DERIVED     | b2         | ref    | F_LU_ID       | F_LU_ID | 8       | const |    0 |                         |
+----+-------------+------------+--------+---------------+---------+---------+-------+------+-------------------------+

Comparar con EXPLAIN informe para la solución dada por @martin clayton:

+----+--------------+---------------------+--------+---------------+---------+---------+------+------+---------------------------------+
| id | select_type  | table               | type   | possible_keys | key     | key_len | ref  | rows | Extra                           |
+----+--------------+---------------------+--------+---------------+---------+---------+------+------+---------------------------------+
|  1 | PRIMARY      | lu                  | system | PRIMARY,ID    | NULL    | NULL    | NULL |    1 |                                 |
|  1 | PRIMARY      | <derived2>          | ALL    | NULL          | NULL    | NULL    | NULL |    4 | Using where                     |
|  2 | DERIVED      | <derived3>          | ALL    | NULL          | NULL    | NULL    | NULL |    4 | Using temporary; Using filesort |
|  2 | DERIVED      | <derived5>          | ALL    | NULL          | NULL    | NULL    | NULL |    4 | Using where; Using join buffer  |
|  5 | DERIVED      | LettingUnitBookings | ALL    | NULL          | NULL    | NULL    | NULL |    3 |                                 |
|  6 | UNION        | LettingUnitBookings | index  | NULL          | F_LU_ID | 8       | NULL |    3 | Using index                     |
| NULL | UNION RESULT | <union5,6>          | ALL    | NULL          | NULL    | NULL    | NULL | NULL |                                 |
|  3 | DERIVED      | LettingUnitBookings | ALL    | NULL          | NULL    | NULL    | NULL |    3 |                                 |
|  4 | UNION        | LettingUnitBookings | index  | NULL          | F_LU_ID | 8       | NULL |    3 | Using index                     |
| NULL | UNION RESULT | <union3,4>          | ALL    | NULL          | NULL    | NULL    | NULL | NULL |                                 |
+----+--------------+---------------------+--------+---------------+---------+---------+------+------+---------------------------------+