MEWTWO
One of the more infamous visual bloopers in the episode "Mystery at the Lighthouse" is the Mewtwo on the door. To be a little more specific, on the door to Bill's lighthouse are a number of different Pokémon carvings. Some of these are completely ordinary Pokémon, and others are legendaries. One of the latter, if a viewer looks at the door just as it's opening (although it's also viewable just after Ash hits the doorbell), is a clearly visible carving of Mewtwo.
How's this a problem? Well, later on in the series (more specifically, in the first movie), it's revealed that no one knew that Mewtwo existed except for Team Rocket scientists directly involved with the project, higher-ranked Rocket officials, and Giovanni himself.
So, what's this mean for Bill? If we view this from the perspective of someone within the Pokémon world, this would imply that Bill is a member of Team Rocket. After all, no one else knew about its existence, so how could Bill have it on his door?
However, there's a few other things to keep in mind.
1. First and foremost, "Mystery at the Lighthouse" is only episode thirteen. The movie didn't come out until over a year later. (It debuted in July 1998, whereas "Masaki no Toudai" first aired in June 1997.) While, sure, there might have been planning in the process, it's more unlikely that the final details of the movie were completely hammered out and revealed to the animators by the time Bill's episode was in development. Remember, in game canon (which the anime is based on, at the risk of stating the obvious), it wasn't originally clear who created Mewtwo. Moreover, considering Mewtwo has a Pokédex entry, it was probably assumed that it's plausible Bill could have easily learned about it through other means.
2. The lighthouse is old, and its keeper is young. While it's possible Bill could have rennovated it, there's also the possibility that the doors were already there by the time he acquired the lighthouse. This begs the question of who might have had the Mewtwo carved into the door and who the original owner of the lighthouse might have been.
3. Even if we do assume that it was Bill who had the Mewtwo carved into his door, the next question is why would he do it? Putting an unknown Pokémon in full view of visitors would put him at risk if he was an active member because people might ask questions about that massive carving of the random unknown creature. Is he a former Rocket who defected and is trying to send a message to trainers? Is he secretly a member of the Global Police on a mission to stop Team Rocket? Is he psychic? Or, even more simply, did he hear a description of Mewtwo from someone else and just thought it was another legendary?
4. Also keep in mind Bill's age. While it's not given, he's described as being young by Professor Oak, and he looks about the same age as Brock. The project that created Mewtwo had probably been going on for years unless some accelerated growth was going on. It's also entirely possible that Bill might have been too young to work on any team that would have produced Mewtwo.
5. For that matter, also keep in mind Bill's primary focus. In the anime, although he's a general Pokémon expert, his primary field as a researcher is Pokémon behavior, not genetics. Beyond that, although he's intelligent, the anime makes no effort to show us that he's also an inventor. In other words, he doesn't seem to have the skills Team Rocket would have wanted for a genetics project. So, he'd really have no reason for being hired by them (for that particular project, anyway) in the first place.
So, what theories can we walk away with? It's hard to say and probably best left up to headcanon. However, although the subject can generate a lot of fanfic plot bunnies (And we encourage you to pursue them if you'd like!), Keeper's chief researcher personally believes that the Mewtwo on the door is most likely exactly what it looks like: a visual blooper that's hilarious in hindsight.