I understand that Naruto's half of the nine tails had been befriended by Naruto but why did Minato's nine tails help? Wouldn't it make more sense that he would've resisted and didn't let Minato use chakra mode or tailed beast mode?
Dark Kurama (the half sealed within Minato) has a good reason to help them. His very existence is under threat by Uchiha Obito (and later Uchiha Madara) Eternal Tsukuyomi. If Obito's team win, he will be sealed into the Gedo Statue and returned to being a part of the Juubi. Regardless of whether the tug-of-war between Dark Kurama and Minato happened or not, Dark Kurama would still help Naruto's side so that at least he can still be him.
With the tug-of-war taking place, he is even more willing to help since he's been tamed by Minato.
It was explained that Minato mastered the tailed beast in the tug-of-war of chakra war and began controlling it just like Naruto did. He gained the same chakra cloak like Naruto had in the 4th Shinobi World War.
He mastered controlling him and also dark Kurama had no reason to not help his other half since his existence was under threat of getting back to being a part of the juubi.
Kurama's other half is inside Naruto and moreover if obito captures Kurama he will be trapped in the gedo statue and Naruto has gained Kurama's respect by treating him as a friend.