Signature kick or not, Ayanokouji will still have a chance. This fight should be very close. Against Ibuki, Ayanokouji could
analytically predict and dodge all of her round kicks, which are very much similar to Mikey's signature kicks. In fact, Mikey is a much easier opponent than Ibuki as he has never shown an ability to repeatedly kick his opponents on the same level as Ibuki (unsure about the final arc till it's animated). Takemichi said this:
Mikey takes a while between his two kicks. A person like Ayanokouji, when determine to win, can pull off crazy stuff as a kid. Not to mention that Ayanokouji is not only aware about lethal attacks, but non-lethal attacks as well. He was able to avoid orange juice flying at him in the novel:
His white room training should make this a skill-stomp already. Not only is he trained for like more than 10 hours a day for 10 years everyday, but also he had level 10 training at the age of below 10, a level normal human should not even think of reaching in their lifetime.
Against Albert, he was able to adapt to the fact that his attacks weren't being effective due to Albert's crazy endurance, so Ayanokouji literally hit his solar plexus and took him down in a few hit. He mentioned taking down Shiba in a single hit and Tsukishiro in a matter of blows. Ayanokouji, with his immense knowledge and much higher battle IQ than Mikey, might even pull it off.
Also, I didn't that this is Kid Mikey and Kid Ayanokouji and Ayanokouji here would have higher stamina than Mikey, so, my own belief is that Ayanokouji would pull it off.
Yes, him being able to avoid all attacks is a narrative by Ryuuen, but so far, it hasn't been proven false. Even when being drained of stamina, Ayanokouji avoided all of Nanase's attacks, and after that, all of Tsukishiro and Shiba's attacks. Even if Mikey's dark impulse awakens and he is determined to kill Ayanokouji, we can easily say that Ayanokouji's motivation to kill Mikey should be much higher (as he should be ordered to kill Mikey, and this Ayanokouji has orders above everything). Ayanokouji has crazy adaptability which allows him to absorb enemy techniques and modify them to a better extent, so, if we see this, Ayanokouji should be able to do a roundhouse kick better than Mikey if he sees him perform one.
AP: Mikey
Speed: Equalized
Battle Intelligence: Ayanokouji
Martial Arts & Skills: Ayanokouji
I believe Ayanokouji would hit Mikey in his vital organs and win this fight very easily. Switching my vote to Ayanokouji until someone convinces me that this wouldn't happen.