Well, whatever I need to be doing to make it update faster, or even at all, I didn't do this run. Started a new run. Kept the Zalian defense tool equipped the entire time. Used a standard ranged deck (Dive for Cover, Single Shot, Short Burst) this time. I won 12 fights today, one against two opponents at once. But then I ran out of energy before it upgraded. If it's still using a per-day counter, I think I'm going to have to give up, since I'm planning to go for the Closed System conduct this run and won't have the turngen to get many fights each day, much less win them with a Zalian thingy in its base form.
Maybe you need to switch between melee and ranged, or use both in the same chain or something to get it to update? Seems like you need to do something weird, because just using it in the normal course of fighting is definitely not working for me.