Here's what I came up with so far. Not what I originally had in mind with my "dream movelist", but that was a while back and I can't remember everything. Also, I changed some of the stuff you had around. Don't be mad. 
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Unique Moves:
Roundhouse - f + HK
Axel does a high roundhouse kick. Standing in front of the opponent knocks them down, while using it as an anti-air leaves the opponent in a juggle state that allows you to combo them.
Shovel Hook - f + HP
Axel punches the opponent at a 45 degree angle.
Backhand Punch - close HP (does this count? lol)
Axel backhands and smacks the opponent in the face with his knuckles.
Special Moves:
-Grand Upper - F, D, DF + P
Axel does his iconic uppercut attack. Goes further and does more hit with each strength.
-Dragon Wing - D, DF, F + P
Axel performs a powerful circular punch that creates a flaming dragon. Eats 1-hit projectiles like Bison's Psycho Crusher. EX version can take 2 hits.
-Fuze Counter - D, DB, B + P (could also be D, D + P)
Axel gets in a ready position and counters any physical attack that hits him. LP counters crouching attacks and has him kick the opponent in the head, HP counters standing and jumping attacks and has him headbutt the opponent, EX is the same as HP but instead has him pull out a wooden plank/steel pipe/bat and knock the opponent away and does more damage.
-Mad Kicks - D, DF, F + K
Axel moves forward a little and performs two quick kick. The HK version knocks the opponent down, and the EX version makes the opponent bounce off the wall for juggles.
-Going Straight - D, DB, B + K (
Axel dashes forward and connects with one of three attacks. LP is a rushing hook that doesn't have much range, HP hits with a wooden plank/bat/pipe, EX does a upward slash with the katana.
Level 1 Supers:
-Dragon Smash - D,DF,F,S
Axel hits his opponent with a flurry of strikes, and then finishes them with a "Tiger Uppercut." First hit must connect and not be blocked, otherwise the super doesn't happen. (Preferrably have him stand in place when the super is executed for accuracy, or can be a rushing for more usage.)
-Grand Hook - F,D,DF,S
Axel does a spinning flame punch along the ground that ends with an extremely powerful, multi-hit version of Grand Upper.
Level 3 Supers:
-Calling All Cars - D,DB,B,S
Could go either one of two ways. Either go with your original idea where the cop car just rolls in, or make it a command grab super where he throws the opponent to the ground, THEN the cop car rolls in and blows shit up. For fun, have the car being driven by Adam Hunter.
Target Combo: LP, LP, HP, LK, LK (does his trademark SOR combo)
As you can see, I added some stuff from some of the enemies/bosses in as well. Give me some critique on this, as I would LOVE to add/modify/replace some moves, since this wasn't that well thought out as I hoped it would be.
Also, yes, I named some of the moves on the track names. I hope you do the same.
