I don't think it's necessarily anything to do with exams.
Just that different teachers teach with different styles. If you have two teachers then they both teach a different style which can be confusing.
Dh swapped teachers after grade 4/5 because he started with a teacher who could only teach up to that level. He remembers changing teachers as being quite challenging, although he is very musical. For that reason he was very keen when we were looking for a teacher for dd1 that they would be able to take her all the way up (if necessary).
I don't understand why you weren't upfront with the second teacher at the beginning. Surely you'd have said that you wanted a "second teacher"?
At any rate the exams do typically one jazzy piece, so they should be learning different styles with one teacher. Dd1 is classical in style, as I think her teacher is naturally, but she is currently doing some jazz pieces in her lesson, and playing some Queen songs for fun. She doesn't need a separate teacher to teach other styles.
You do actually come across as quite pushy. Why isn't one lesson enough? He should be able to practice well from a lesson a week. You can get other books for him to use in practice if he wants to do more.
Why don't you see if there's a different instrument he's interested in for the second lesson?
In the same way my dc's ballet teacher likes to know if a child is having dance lessons with a different teacher. She advises not to, simply because of the contrast in style. Nothing to do with entering for grades or anything. Dd2 did a social class for a time with a different teacher when she was quite little, and you could see at times where she was confused in her dancing.