In order to maintain the same teams from game to game in a format with randomly generated teams, you'd likely have to force the RNG seeds used for generating teams in game 1 to be the same throughout the best-of-3. Either that or you'd have to completely bypass the team generator for the randomized format and feed it in the team it generated previously. We currently don't have a mechanism for anything like this.
It sounds to me like there is no actual format or tournament being used for this, but presumably it's just a custom challenge with rules like Adjust Level Down = 50, Team Preview, and Picked Team Size = 4. I am hesitant to approve a feature for a niche custom challenge format like this. I suppose BSS Factory could theoretically want it though, and if there was ever a VGC Battle Factory it would need this support.
I'll approve it for now. For the developer: the default behavior should be the current behavior (bo3 randbats do not keep the same teams). If a player wants to opt-in, they should add another custom rule that only works if a Best-of is set and the format uses randomized teams.