Does every melee weapon break or is it just swords and daggers and stuff like that?

The problem seems to be with the iron, so staves are OK, and magic weapons are unaffected.

Yup it's part of the storyline, the iron problem.

Getting a magic weapon will prevent it from breaking, so try get something that's +1 or has a fancy name, until then you can just carry around a spare weapon.

Luckily the chance is small, so a spare metal weapon should do. If you want to make the iron crisis seem more reasonable, use the Breakable Non-Magic Armor, Shields and Helms component of BG1 Tweaks for vanilla BG1, or the same component in BG2 Tweaks for the TUTU conversion (the BGT conversion has it by default).