He could get lost. Bugs Bunny is used to his familiar territory in the forest or the carrot fields. If he got lost in a strange place, like a big city or a spooky swamp, it would be a bad time. He'd have to figure out how to get back while avoiding new threats.
I'm not sure specifically which 'bugs bunny bad time story' you are referring to. There could be many adventures of Bugs Bunny that might involve some tough situations. Maybe it's about the time he got chased by Elmer Fudd more intensely than usual or when he faced some particularly tricky traps in the carrot patch.
A bad time story related to bugs could be when you are on a date in a park. You've set up a nice picnic, and suddenly, a bunch of ants start climbing all over the food. Your partner is grossed out, and the romantic mood is completely ruined. All you can think about is getting away from those bugs instead of enjoying the date.
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One possibility is that their baby would have Bugs' famous carrot - loving trait. It could be shown as a cute little addition to the Looney Tunes gang, with the other characters being both protective and amused by the baby. For instance, Tweety might tweet helpful advice to the little one.
Once, I was about to have an important outdoor party. Just as I was setting up the tables and chairs, bugs started swarming everywhere. There were mosquitoes biting people and little black bugs getting into the food. It was a really bad time because everyone was uncomfortable and some guests even left early because of those pesky bugs.
Maybe in the fanfiction, Bugs Bunny is a summon of Naruto. Naruto discovers a new type of summoning scroll that somehow connects to the Looney Tunes world. Bugs Bunny's abilities like his super speed (which in the context of the Naruto world could be seen as a unique form of chakra - enhanced movement) and his ability to talk his way out of trouble would be very useful to Naruto in his missions.
One main event is his constant run - ins with Elmer Fudd. Their chases are iconic. Another is his encounters with other Looney Tunes characters like Daffy Duck. Bugs often outsmarts them all.
Maybe he could read 'Little Red Riding Hood'. Bugs Bunny could add his own comical twists to the story, like when the wolf tries to trick Little Red. He might make the wolf stutter or say something silly. Also, 'The Three Little Pigs' would be a great choice. Bugs Bunny could really ham it up (pun intended) when he tells the story of the pigs building their houses and outsmarting the big bad wolf.
The plot might involve Bugs Bunny rescuing Penelope from some sort of danger. This heroic act makes Penelope see Bugs in a new light. They then bond over their shared experiences and gradually fall in love. For example, if Penelope was trapped by a mean hunter, Bugs could use his wits to free her and from that moment on, their relationship would develop.
In a bunny pajamas Christmas story, there might be a lonely old man who receives a pair of bunny pajamas as a Christmas present from a kind neighbor. He puts them on and suddenly feels a sense of joy and youthfulness. As he goes out for a walk in his bunny pajamas, he meets other people who are also in high spirits because of Christmas. They all share stories and treats, and the old man has the best Christmas he has had in years.