Converting your license to the happy islands is an easy process. Just organise it with the Civil Aviation Department- an open book air law exam based off the British Overseas Territories Air Navigation Orders (or something like that, it's been a few years since I was there, things may have changed though.)
As for finding an aircraft there to fly, that is where you are likely to hit a wall. There were no private aircraft based there in my time, only things that ferried through.
Solomon Airlines won't give up a plane for your private use, but you could of course charter them to do some runs for you.
Your best bet is probably taking a ZK machine over. If it works out and is long term maybe re-register it locally, but maintenance is going to be a massive hurdle.