I think Banjo-Kazooie would be a floaty/flying projectile zoning character. While they would have several options for close-quarters combat, their main forte would be zoning people with eggs.
Their neutral B would cause Kazooie to shoot eggs in an arc-like fashion and their down+B would be called "stop 'n' swap" where they swap to a different type of egg ammunition.
Blue Eggs - standard projectile, can fire up to three at once. 2% damage per egg, does the same knockback as a jab.
Fire Eggs - increased knockback and damage, but reduced firing rate. 7% per egg, does the same knockback as DK's f-tilt.
Grenade Eggs - explosive projectile, high killing power. 15% damage per egg. Can also blow up and damage/kill Banjo-Kazooie if they're not careful.
Clockwork Kazooie Eggs - hatches a miniature, fast-moving clockwork Kazooie which is controlled by the player. Banjo & Kazooie are unresponsive to controls while the clockwork Kazooie is out. Damaging explosive similar to Bowser Jr.'s mechakoopas, crossed over with Snake's remote-controlled missile from Brawl. 9% damage upon explosion, decent killing power like Din's Fire.
Their side+B would be the wonderwing, which can work as a catch-all barrier move. Banjo continuously runs while it's active. It has high knockback on contact with an opponent, and is invulnerable to all projectiles. Can be blocked and grabbed out of shield, and also has high recovery.
Up+B would be their flight pad takeoff, which can be done in the air. After launching upwards very high, Kazooie will slowly flap her wings as they glide to the ground. The opening part of this move counts as an attack.
Most of their other moves would be relegated to the A button. Rat-a-tat rap would be their forward tilt. Breegull Bash would be their down-smash, with Banjo comedically smashing Kazooie on the ground on both sides of him. Beak Barge would be their forward Smash, and their down-air would be beak buster, which can be held and charged into a Bill Drill.
I don't like the idea of the Jinjonator being their Final Smash when it could be something much more prevalent in the games. Their Final Smash would involve Mumbo Jumbo appearing and transforming them into something, like a washing machine or a giant T-Rex.