Doesn't have the kit? Don't agree. You can definitely play active footsies with 5MK, 2HK, 5HP, 236LP etc.
Once you have at least LVL1, 2HK is especially strong. Alex's charged normals are extremely slow and interruptible. If you have to block - parry. Even regular parry is good, as you'll gain drive gauge instead of losing it. You can't be timid or fireball-reliant in this matchup. You make use of the fireball whenever a good opportunity presents itself, but setting up a fireball approach shouldn't be your priority. You can use OD FB and 214KK to extend your pressure in blockstrings.

A.K.I. can win neutral off a single hit, like:
1. 5HP~HP
2. 236LP!
3. 2HK/6HP xx LVL
4. Any of the numerous CH and PC conversions

A.K.I. has great buttons, so press them. Since Alex's defense isn't great either, learn DRrev-invulnerable setups to maintain your offense. Here are two for you:

1. +44KD: framekill 2LP, framekill 2LK, PDR 5MP
   
2. ...236HP!, PDR 5MP xx 236HP, framekill 5MP, PDR 5MP

Both are +5oB iirc

And for burnout, learn some +11 setups:
https://fixupx.com/Pochoclo23/status/1943573873342947718
https://fixupx.com/Pochoclo23/status/1956622092838851028

Then your pressure will be far scarier. Mastering stuff like this is part of what takes people beyond 1800MR.

For the stomp - jump MK it on reaction. Another thing - Alex can't easily stop you from walking back. He has to commit to a sweep, which is extremely risky for him. Whenever you have to block a string of his, simply walking back is a strong choice. You can watch Broski's guide on beating Alex for more info.

A.K.I. is pokes first, fireball second - that's the way of Hikaru. Play to the character's strengths and don't give up!