Referring to the little book on the Bembridge branch by Peter A. Harding, informs me that essentially, Bembridge branch trains were always hauled by the loco in each direction; though the line would seem a good candidate for push-and-pull working. The Bembridge branch did experience a short emergency-measure spell of push-and-pull working, powered by a "Terrier" 0-6-0T, in spring 1936 while the "quasi-turntable" at Bembridge station, which aided in running-round, was being enlarged.
Harding implies that the most prominent push-and-pull scene on the IOW was the Ventnor West branch: in later SR, and BR, days. This tallies with a photograph which I have seen of a push-and-pull train, 02-worked, on that branch, shortly before its closure in 1952.