It would probably be difficult to dry-hire an aircraft from a commercial charter or airline operator, due to licencing and insurance requirements. I am unaware of any flying school or club there with aircraft available for hire.
However, I don't see why you could not fly privately within the Solomons using a ZK registered aircraft. It would have to be short term, because most countries only allow a foreign aircraft in on a temporary basis before import duties etc have to be paid, but if you were coming and going to somewhere else in the region that could be do-able. You could possibly get 6 months then fly it to PNG or Vanuatu for some maintenance, then get another 6 months before someone in Customs stuck their hand out.
Another issue could be availability of avgas, but for as long as there are Islanders there, some must be coming in to the country. I hate to imagine at what cost, because the quantities would be limited.
But like most things in aviation - if you chuck enough money at it, it CAN be done.