In terms of base power, Bell would probably surpass Ais on paper once he reaches level 5. But Ais is one of the best fighter. So it should take high level 5 to fight Ais on par as long as they dont use magic. Argonaut needs charging so, in order to use it against Ais, Bell has to charge up before he enters the battle.

Will Bell get stronger than AIS?

No they aren't. A normal adventurer doesn't get to the limit. They usually reach Ds and Cs, B with luck, so the difference between Bell and most of them is even more.

Does Bell ever fight AIS?

During the Xenos arc, Ais and Bell fought when he tried to protect Wiene from her. Despite this, they were able to make up after it.