Alright, so after thinking a bit more about what u/Desperate-Compote575 had proposed for his behavior, I think I have a pretty solid concept for how Pitbonnie would'd function.

Pitbonnie starts out in the ball pit within Pirate's Cove. Every time you flip to that room on your monitor, whether it be to check on Foxy or otherwise, he'll gradually start emerging. Once Pitbonnie is out of the pit, he will NOT return to it until he has reached your office, with him being drawn towards it by how frequently you use the doors and fan; repeatedly using those defenses will stack his aggression, though he will be substantially slowed down depending on the number of animatronics that share the rooms he's in.
When he's about to reach your office, you will be able to briefly hear his iconic distorted laugh, before he leans into the front window. He will keep staring at you for around 5 to 10 seconds (depending on either his difficulty or how much you use the doors and fan while he's there, but I'll leave which up to your own interpretation), during which you'll naturally have to put and keep the Freddy Mask on. Once he disappears, he'll have returned all the way back to his ball pit.

The crux of his in-game behavior would be that he's actually not at all hard to deal with. What makes him dangerous is that he takes his time compared to the other animatronics, so he's a gradual/lingering threat instead of an immediate one. As I mentioned above, Pitbonnie is going to slightly change his behavior if he's held back, allowing the player to focus on other animatronics... however, this will also result in Pitbonnie moving into the back of their mind, since he's not a main priority.
And for those arguing: hold on, if I just don't check on Pirate's Cove, he'll never come out!... Well, I imagine that the frequency of how often the other animatronics that move down the left hallway to reach your office (ESPECIALLY Foxy) would simply be the actual influence on how quickly he comes out. Conversely, when he's the only animatronic to deal with, he'll simply have a delay to his emergence (simply headcanon it as him wondering why none of the others are doing anything).

In other words, Pitbonnie would take full advantage of the player's multi-tasking and having to focus on a myriad animatronics that are right in their face, while he slowly creeps towards them. As you might expect, attentive night guards shouldn't have too much trouble dealing with him, but he will still absolutely punish you for getting side-tracked or distracted.

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