Firstly, lets make a correction. Naruto learned the Summoning Jutsu, not before the exam, but only before the third round, prior to his fight with Neji. So he would have only had one opportunity to use the Jutsu during the exams, during his fight with Neji.
It is never EXPLICITLY stated, as far as I know, why Naruto didn't use Summoning Jutsu to summon Gamabunta during the fight, but let me give you a different perspective. Why would he want to summon Gamabunta in his fight against Neji?
- He had a personal reason to fight Neji. He probably wanted to beat Neji on his own, 1v1, to prove his point to Neji. Probably wouldn't feel great to beat Neji because he had a giant frog step on him.
- It would have been impractical. Gamabunta is HUGE. Gamabunta would probably have filled the entirety of the stadium they fought in. Even if Gamabunta did fight, he sure as hell wouldn't have been able to do so without destroying the stadium to hit a single tiny human target.
- Like you mentioned, Gamabunta probably wouldn't have even wanted to fight for Naruto. The only reason he fought with Naruto against Gaara later is because he knew about the One Tail Beast and Gamakichi convinced him to.
- Naruto had literally only used the technique once successfully in his training with Jiraiya, under threat of falling to his death. He probably didn't have confidence he could use it again. This is evidenced by the fact that in his fight with Gaara later, he only successfully summoned Gamabunta again because he was about to die, failing the first time by summoning Gamakichi. Naruto isn't able to reliably summon Gamabunta for quite a while in the series, summoning Gamakichi and Gamatatsu on multiple occasions.