I have had a look through my books for the period and I realise why you are calling for information. The entire class only recieves a couple of lines in Janes. The USN called them ATR (Auxiliary, Tug, Rescue) and merely added a B prefix for the units allocated to Britain.
There is more information in the second edition of a book called "Warships of World War II", by H T Lenton & J J Colledge, published by Ian Allan in 1973. The second edition had a much expanded coverage of suport ships and provides the following information on HMS Director and her three sisters, Emulous, Freedom and Justice: -
Displacement: 1,360 tons
Dimensions: 147 (bp) 165 (oa) X 33.5 X 15 feet
Machinery: Single shaft; Triple expansion steam engine; 1,875 IHP = 12 knots.
Armament: 1 X 3 inch; 2 X 20 mm single mountings.
HMS Director was launched on 13. 7. 1943 and returned to the USN on 23. 4. 1946. Rescue tugs were employed in assisting damaged merchant ships back to port.
Page 369 of the Ian Allan book has a large and clear photo of HMS Director, which was provided by the Imperial War Museum. In view of the entries in the current thread about posting illustrations from books and that IWM photos are Crown Property, I do not intend to scan it. IWM will sell you a copy, if you contact them at [email protected]
Hope this helps