It certainly predates 20th Century councillors...
Harley Shute can be found marked on Ordnance Survey maps of the area from at least early 19th Century to present day.
The National Archives makes reference to a 16th Century manuscript held by East Sussex Record Office regarding the area in question.
AMS5695/2 - An 18 year lease dated 15 July 1557 -
Rent 110s p.a for � the demesne of the manor of Harele (in Hollington, St Leonards and Bulverhythe).
John Sandell of Hollington by his will, proved May 1557, bequeathed his farm of Harlye to J H and J P.
J P covenants to repair hedges, banks etc. J H covenants to repay J P 16s. 8d. for each remaining year of the lease if J P is legally evicted from the property. Certain rights are also reserved to the heirs of Dame Mary Fitton.
Present at sealing: Richard Byshopp, jurat; Richard Russell, Edmund Smyth, Thomas Bryan
J H = John Hemelyng of Hastings
J P = John Poppe of Hollington
Dame Mary Fitton held the Manor of Filsham, a manor dating back to Saxon times. The manor house was rebuilt in 1682 and part of this remains on present day Harley Shute Road.
http://www.nationalarchives.gov.uk/A2A/records .aspx?cat=179-amsj&cid=54-2#54-2
It also strikes me as unlikely that a road named after a Harley-Shute would have a side road (Harley Way) utilising only half the surname.