Bonin would have had to realize that the plane is in alternate mode in the first place in order to react correctly. It probably never crossed his mind since he never trained in it, and conversely if he had trained in it, he may have been more likely to at least check.
But to me the bigger problem seems to be that such an important change in the plane's behaviour could happen without anyone noticing. I'd consider the mode to be something the pilot must be made aware of, not something he has to deduce from the fact that the airspeed isn't available.
Perhaps the mode is shown prominently and the pilots just didn't notice it in their state of panic. Making it more prominent probably leads right into an insane arms race - the stall warning was as prominent as anything can be and still got ignored.
I don't envy the person who has to design a airliner cockpit's user interface and decide which of a hundred potentially vital pieces of information should be displayed how.
Bonin had to have known that the plane was in alternate law.
In the flight recorder log, at 2h10m05s, there was an audible "cavalry charge" alarm that indicated to everybody in the cockpit that the autopilot was disconnecting (plus message at the same time on the ECAM).
Then, on the ECAM message console 1 second later, the message "F/CTL ALTN LAW (PROT LOST)" was displayed: alternate law, protection lost. At the same time, Bonin said "I have the controls", which to me indicates that he knew that the autopilot was off and that alternate law was engaged.
A mode switch as important as this should be very visible. An idea: put a two-color led under every indicator light and switch to the other color when the plane falls into alternate law. This should make it sufficiently clear that something very notable has just happened.
Something like that was my initial thought. However, you have to remember that there is massive amount of very notable things going on and data shown. As brazzy pointed out, the stall warning was as prominent as possible and still was pretty much completely ignored. I don't think they would've noticed any amount of leds in that point.