My problem with Bruce isn't him firing her over this incident, it's the whole weird set up of his use of Robin. It seems to be using Steph (to manipulate Tim?) and frankly disrespectful to Robin the position.
Robins disobey sometimes. It's part of their job. As Dick says to Tim once, Robin needs to anticipate Batman's orders, which means he has to know what to do on his own, and that means sometimes he's going to have to choose to do what s/he thinks is right or go with Bruce's last order. And even if s/he's wrong in a particular instance, s/he needs to be able to make that choice sometimes. Bruce doesn't train sidekicks, he trains independent superheroes capable of leading other teams into dangerous situation before they're old enough to buy liquor. Sometimes Robin gets new information that overrides the order, or can see that Bruce is not giving good orders. Despite the shortpants, Robin is not a position without respect.
I have no trouble with this instance being different because I think it is. Stephanie is, imo, being written here as somebody who isn't a good fit for the job tempermentally in ways that Tim, Dick and JasonI aren't written in stories where they disobey an order. I don't have a problem with Batman rejecting somebody for the position, especially since Jason.
But my problem is Batman seems to already know perfectly well why she's not right for the job and is just using her and the role of Robin for some weird mindgame of his own with Tim. Which imo seems to be the way Steph so often gets treated by Bruce. Like, it's not like he gives her some reason why he thinks she shouldn't be Spoiler, he just continues to make it about "because I say so" which is guaranteed to make her continue--almost as if he wants her to do that. And he does like using her in the ways he thinks she's useful. It's just sees more like a tool for him where Tim and Dick are more on his level. So much weirdly wrong in the Batman/Spoiler relationship (another reason I hated the retcon of Barbara's story to try to replay it).