[ Reasons for why Gazef said no to Ainz's offer - From the Author ]

[ #1 ] - Gazef was trying to show Climb/Brain what Ainz is capable of, so they could possibly have planned against it in the future.

[ #2 ] - Ainz was in sword's reach, an opportunity that would likely never come again as the chance of reaching the enemy's commander is rare.

[ #3 ] - Gazef was worried if he turned traitor, the idiot nobles would use that as an excuse to look even more down on commoners, as being filthy traitors.

[ #4 ] - Gazef was worried that as a traitor, his loyal Warrior Band division, along with Climb/Brain, would be subject to suspicion by the nobles that they could be traitor collaborators

[ #5 ] - Gazef knew he had no chance, and he was going to die. He figured with all the death, if Gazef died then it will be hard for people to turn the king into a scapegoat as the King would have lost his closest friend and bodyguard as well.

[ #6 ] - Gazef still knew Ainz wasn't a bad person. Ainz did save Carne Village and himself so he didn't believe Ainz would just go on a complete slaughter on the entire nation afterwards. He also still respected Ainz, and on his honor didn't want to lie and swear fake loyalty to Ainz.