Despite having a connection to the team in the comic iteration, Beast Boy has had no role in Doom Patrol since its start. While Titans was a huge series to join the DC TV line-up, the drama ended up helping launch another team from panel to screen. In Titans season 1, fans got to meet the Doom Patrol, a team that was fairly obscure to the mainstream audience. When Ryan Potter’s Gar Logan a.k.a. Beast Boy was getting his proper debut episode, the show took the opportunity to introduce the team that he’s affiliated with in the DC mythology. Even though it's a Titans spinoff, Doom Patrol hasn't even alluded or referenced Gar having anything to do with the team.
When the team's official line-up was announced, all the classic members were attached, from Niles Caulder, Elasti-Woman, Negative Man, to Crazy Jane and Robotman. But instead of Beast Boy, the spinoff had a different character included that, initially, was an odd surprise to some. Despite not having any ties to the team in the comics whatsoever, Doom Patrol was going to include Victor Stone a.k.a. Cyborg who is actually a member of the Teen Titans. Joivan Wade ended up being cast as the latest live-action iteration of Cyborg and has been part of Doom Patrol for two whole seasons now.
But now that Doom Patrol has established its world for at least two seasons, why hasn't Beast Boy appeared at least once? The reason why Cyborg was likely due to the fact that Vic has been one of the big Justice League members since The New 52, which would bring some familiarity to fans who would be exploring Doom Patrol. Since the team was still very new to most audience members, having Cyborg as one of the principal characters certainly helped. Given that Cyborg had only gained some recognition after appearing in the 2017 version of Justice League, that probably helped raise his status in the media compared to Beast Boy who had only been in animated shows at that point.
However, there is also a lot of logistical reason why Beast Boy has yet to appear on Doom Patrol. The first keynote is that despite being a spinoff from Titans, Doom Patrol ended up going in a different direction with its continuity. Rather than taking place in the same universe, Doom Patrol was retroactively set on its own Earth where Niles had never met Gar, hence he had never brought him on to the team. The Arrowverse crossover event "Crisis on Infinite Earths" became a second confirmation as Doom Patrol is set on Earth-21 while Titans take place on Earth-9 in the post-Crisis world.
But theoretically, even if Doom Patrol were to have Potter play the Earth-21 version of Beast Boy, that could still be complicated in the long run. Given that he is a series regular on Titans, it would probably be tricky to make shooting schedules work out. Titans films in Toronto, Canada, whereas Doom Patrol is actually filming in Atlanta, Georgia, so it's not like the Arrowverse shows that mostly all film in Vancouver. At this point, if Beast Boy ever does show up on Doom Patrol, it would likely be for a small number of episodes or maybe just a cameo.