One idea could be that the misfortunes from 'No Such Luck' somehow cross over into an alternate universe in the crossover. For example, Lincoln's bad luck could start affecting the other characters from the crossover world in strange ways, like causing their gadgets to malfunction.
Another plot might involve a character from the 'bad luck' side of the crossover coming to the Loud House world and bringing their bad luck with them. This could disrupt the normal family dynamics. The Loud siblings would then have to figure out how to get rid of the bad luck and restore order. They might try all sorts of wacky plans, like using their own unique skills and talents in a new way to combat the bad luck.
One idea could be a crossover with another popular animated show. For example, the characters from 'Loud House' could end up in a different universe where they have to use their unique skills from 'No Such Luck' to survive. Maybe they meet characters who have similar family dynamics but different challenges.
How about a character from a classic fairy tale? For instance, Cinderella could be accidentally brought into the 'The Loud House' world during the crossover. Her presence could contrast with the chaos of Lincoln's bad luck. She might try to use her positive attitude and her experience with hardships to help Lincoln deal with his situation.
One common theme is family. Since the show is centered around the Loud family, fanfictions often explore different family dynamics. For example, how the siblings react to Lincoln's supposed 'bad luck' in different ways than in the original episode.
For Lincoln in this crossover, you can focus on his resilience from 'No Such Luck'. Despite all the bad luck he had, he still tried to keep a positive attitude. In the crossover, put him in difficult situations where he has to use that same tenacity. For example, if they cross into a dangerous new world, he could be the one motivating the others.
Leni and a bad - luck character who is always in a hurry to fix things but just makes them worse. Leni's somewhat oblivious but kind - hearted nature could either be a hindrance or a help. They could get into all sorts of misadventures together, like trying to fix a fashion disaster that the bad luck caused, but instead creating an even bigger mess.
One possible fanfiction could be that Lincoln constantly has bad luck in his attempts to get some alone time in the house. For example, every time he plans to have a quiet reading session in his favorite corner, one of his sisters accidentally spills something on him or makes a lot of noise right there. It could also be that his bad luck extends to school, where he always seems to be the one picked on by the teacher for things that are not his fault, like when his classmates play pranks and he gets blamed in the Loud House Lincoln - bad luck scenario.
Perhaps a fanfiction could explore a different dimension where Lincoln's bad luck actually turns out to be a superpower in disguise. He could start to use it to his advantage and save the day in unexpected ways.
You can start by looking on fanfiction websites like FanFiction.net or Archive of Our Own (AO3). Just type in 'Loud House No Such Luck' in the search bar, and you'll get a list of fanfictions related to that episode. Some might be really creative, like those that explore different endings or what could have happened before or after the events in the episode.