Defending champion Marcin Bialoblocki (Britain Velosure Giordana) led a breakaway of 11 riders including Fankhauser over the line, finishing minutes ahead of the main bunch featuring Bevin.
There were a further five categorised climbs today following a brutal stage four, with Bevin and the New Zealand team suffering from tired legs.
Bevin was part of a 20-strong break that established a lead of 1min 20sec over the main field, bridged the gap briefly to the front before being dropped on the descent of the category two climb of Caha, along with 10 other riders.
Bialoblocki accelerated from the remaining group to claim the victory and push Bevin back to 13th place overall, now eight minutes down on the leader.
The New Zealand team will focus on protecting Bevin's lead in the points classification in the remaining three stages as the race moves north to its ultimate finish in Skerries near Dublin.
Bevin is part of the New Zealand team who are using the tour to prepare for endurance track events at the Commonwealth Games.