Why Falkland Islanders' choice governs the sovereignty question

Would dividing the Falkland Islands between Britain and Argentina be a sensible settlement?

No. The islanders should choose their own status, and their stated preference is to remain British.

Dividing East and West Falkland between Britain and Argentina would disregard the islanders' right to decide their political status. Their overwhelmingly stated preference is to remain British, and outside governments should respect it. Continued Argentine pressure makes persuasion harder rather than creating consent. Sovereignty here is a matter of self-determination, rather than a territorial bargain made over residents' heads.