Wyrm Green Dragons have an AC of 40, not counting the Mage Armor that they'll probably have. The dragon ready for combat will probably have an AC in the mid 50's. Even though archer types have crazy to-hit bonuses, they aren't that crazy.
I don't really see how the archer has a chance. Do you think a 10th level deepwood sniper could defeat a 15th level sorcerer that's fairly tough? I don't. If you can't do that, then what business do yo have messing with a Wyrm dragon that's a 15th level sorcerer plus several HP, plus 35 points of natural armor, plus better attacks than a fighter, plus amazing sensory powers better than a rogue's or ranger's.
EDIT: Looking at the archer's attacks, he's screwed, unless he has some trick in mind that prevents him from getting effective archery items. A 10th level archer should have bracers of archery for another +2 to hit and is best using GMW effects on his bow and arrows. Some kind of dex booster might be useful too. All kinds of potions would come in handy. If the archer is to have a chance, he needs all kinds of cheese.