Nobody has ever answered this question satisfactorily.
Catches outside of the field of play are routinely ruled as outs. Why would you think otherwise?
I believe HR because bull pen (in your example) is not in the field of play. If the player catching the ball reaches beyond the yellow home run line and lands in the playing field while still controlling the ball then it’s an out.
If the fielder leaves the field of play, catches a fly ball; the batter is out. Doesn’t matter where. Think of catchers rolling over the dugout fence and falling into the dugout. All runners advance one base based on the ball being out of the field of play. If the fielder is out of bounds and makes the catch(standing a tarp roll or a catcher standing in the dugout making the catch) the ball is dead and no out is recorded.
Some leeway is allowed by the officials in control of the game in the form of player control after the catch. For the most part, if the activity of the catch carries the player out of the field of play, the player only has to prove control of the ball, regardless of where the player ends up. The activity of the catch MUST be initiated within the field of play.
Field rules will define the field of play; whether the bullpen is in play or out of play (almost all MLB fields designate the bullpen OUT-can’t think of one that doesn’t). If the bullpen is “in play” according to field rules, then anything hit into it is a ground-rule double.
The question I have is the ruling regarding a ball that is caught outside the field of play and then control is lost while returning it to the field of play. Home run or ground-rule double? GRD, I think.