This sure is an interesting take! And I would understand it better if you were talking about their relationship in another media form, but you chose Wayne Family Adventures.
Now before I begin, i wanna make it clear that this take is fine, and I'm simply expressing my own opinion, if you have anything to add/discuss with me, please be cordial and polite.
I personally don't agree with this analysis of this media. First off, WFA is not canon, it's barely even based on canon, so there's no reason to believe that any canon (comic) storylines have happened in WFA unless it's confirmed on screen.
Secondly, because WFA is not canon, and doesn't follow a canon storyline, we can think of it as an alternative universe. In this alternative universe, Bruce is a significantly better father/person than he is in many comics, so again, there's no reason to believe their dynamic is the same as comics.
In this alternative universe, Bruce has never once given us any reason to think he treats Tim like this, and Tim hasn't given any reason either. So, there is no prior evidence of Bruce mistreating Tim in an emotional capacity on purpose
Now, onto the actual episode and my analysis/thoughts on this opinion you've stated! Spoilers for Batman: Wayne Family Adventures up to episode 121.
In the episodes leading up to 121, we know the plotline is that Joker has escaped arkham and is currently on the loose, he has already kidnapped Duke. Duke did manage to escape, and is now safe in the manor. Once it is revealed to the batfam, that Joker was the one to kidnap Duke, Bruce calls in for his family to get home and to turn in their masks until Joker is caught and back in Arkham.
Because the kids are notorious for not listening, and because they believe that Bruce is making a mistake, they all band together at Tim's apartment to make a plan of action. Jason has a panic attack and goes out to find Joker, Dick finds him and calms him down, Jason decides to leave the city for a while.
The other batkids all go to several locations to gather evidence/ find clues. During this, Bruce also decides to go out and find clues/evidence.
Tim is not the only person Bruce lashes out to! In many of the episodes regarding this plotline, Bruce is seen lashing out to Alfred, and his other kids as well.
these are the bigger examples, but even when he is not actively yelling at/fighting someone, we see that he is tense and angry and worried.
Now, in episode 121, Bruce has already talked to his children, prohibited them from their masks, and has already had a run in with one of the batkid teams and told them to go home.
What i want to take into account is that Bruce is feeling out of control. Its very clear in the way he is acting/ the way he is drawn, that he is stressed and likely very worried his kids are going to get into trouble and he wont be able to save them (trauma response anyone?)
So by the time he gets INTO ARKHAM (keep in mind that Tim and Steph aren't just anywhere, they are in a "psych ward" that currently houses dangerous unstable criminals capable of murder). Bruce is already at the end of his rope! In his mind, he is constantly trying to keep his family safe and away from the danger, and at every turn he is being disobeyed. In his mind, he's trying to make sure nothing happens to his kids, and they're literally running into the fire headfirst.
So when he sees Tim AND Steph at Arkham, he blows up. I personally don't believe this has anything special to do with Tim. If you were in Bruce's shoes, there's only so much you can take until you fall apart. If it were any other batkid, Bruce would have done the exact same thing because it wasn't about which batkid was there, it was the fact they were there at all.
Ok, now that I think i've made my point, I'm going to address some stuff before anyone comes at me.
1. Bruce was NOT in the right for blowing up at Steph and Tim. At all. In part 3 when i explain his mental state, its not me giving him an excuse to be an asshole to his kids, its just me pointing out that there is no bias in his mind over Tim.
2. Bruce should not have blown up at his kids at all. He should let them help, and trust in them more, which is the lesson i think hes going to learn at the end of this plotline, i just urge you to keep in mind that Bruce has trust issues! Of course hes not going to easily trust his kids to be safe, not because he doesnt think they're capable, but because of the criminals and the horrible things hes seen.
3. Of course Steph and Tim didn't deserve to get yelled at, i want to make it clear that this was an Asshole move TM, and that it never should've happened. But it did happen, it doesn't take away from the fact Bruce in WFA loves all his kids equally, and that he DOES see Tim as his kid, and that he sees Steph as family (not as a kid tho)
Ok, sorry for the super long post, and if i missed any points or if you want me to explain anything better please let me know! If you think i was rude or mean at all in my explanation, let me know! My idea wasn't to be mean or to hate on your opinion, just to have a discussion on different interpretation. If theres anything you think i should change to make my point more clear or to be more polite, let me know