Hi Murray, there are no private single engine aircraft in the Solomon Islands.
There is a VH registered Baron B55 based in Honiara, belongs to a shipping company, not likely to be available for private hire.
Solomon Airlines has a couple of BN2 Islanders, you may be able to charter.
Where do you intend to fly?
Most ALA's are short and natural surface, apart from Honiara, there are some DHC-8 suitable strips.