Agree with Raymond about the SE trades being fairly well set in in May, but Cooktown to Cairns is not too hard even then. Just be prepared to do a lot of tacking between shore and the reef.
(I did Lizard - Cairns back in September in the same sort of conditions in a Farr 1/2 ton)
I would definitely NOT recommend going via TI. That last 300 NM will be a b*tch.
The trip from Cairns to PNG will have a much more favourable wind/sea angles (I'd recommend going the long way via Lizard/2 mile passage
rather than straight out through Trinity passage)