I don't know about ancient duel rules to say whether bringing a sling was permitted. The take is more along the lines of "David wasn't an underdog. If anything, David was the clear favourite to win because of his weapon". Because a sling at the time was a highly effective and deadly weapon which was still regularly used for centuries after the supposed events of that biblical story because of its effectiveness.

Whenever I've read that passage, I've usually considered his underdog-ness to be from him deciding to not wear even the king's fancy armor, and from him being the youngest child in his family. So he looked even more dwarfed by Goliath than he would have had he worn armor, and since he wasn't one of the elder children in his family (despite being a teen or adult), no one in the culture at the time was expecting him to become a hero--much less a king later! But him eschewing the armor in order to keep his agility and range of motion for using a sling makes total sense--I think King Saul was just miffed that his offer of his own personal armor was rebuffed 🤣

EDIT: minor correction

I went to catholic grade school and got lessons in the bible often enough

I have not read the bible in probably 30 years but what I remember is that the fight was more like the sheriff coming to town to knock some heads. Goliath was the monster enforcer who was able to just clear the room. Like movie star brute and shit. He was coming to kick some ass and David was just one of the guys in the right place at the right time and with a nasty sling talent. The sling wasn't really considered a deadly weapon by anyone. David stepped up and one shot the mother fucker in front of EVERYONE