Although 'fish' is probably a better definition of the solution than 'tickle', that parsing doesn't really work (and the clue could then simply have been written as 'Tickle fish'). It is the agent noun associated with the solution (ie someone doing the tickling) which is also the Australian spelling of the name of a fish.
That said, I do think that the grammar of the clue is somewhat questionable - "One doing this" should refer to a gerund (eg 'tickling') rather than a noun or an infinitive verb (eg 'tickle').