Where does the baseball bat two handed weapon drop? I have no clue and any information I find looks outdated.
A quick google search using "dcuo baseball bat" revealed the answer. http://dcuniverseonline.wikia.com/wiki/Baseball_Bat
So don't IF you have already done the Lighthouse mission and have gained entrance to the final mission where you battle Bane you can go back in anytime and simply run the thing again and again until the bat drops. I did the exact same thing with the Monarch Playing Card Co. mission to get the Deathstroke chest to drop. Be warned you WILL need to give it a break between runs to allow the mission to reset so you can't simply walk out and turn around and walk right back in.. The place will still be empty. Here's a trick I use to get things to reset quicker. When you get outside after completing the mission switch characters.... when your character list comes up just log the exact same alt back in and you'll be standing outside the entrance to the Lighthouse and can go right back inside and all the mobs will be respawned. It could take a bit to get the bat BUT the mission itself won't take long .. At Tier 7 you'll fly through mobs to the final battle with Bane. At that point you have your ball bat and no new alt was required at all.
https://forums.daybreakgames.com/dcuo/index.php?threads/baseball-bat.254810/ I got it from New Genesis Now and Happiness Home. It's the 102 style. And trust me, I was happy they dropped finally. As I was originally farming Dox because it was in there, but they removed it when they adjusted the weapon levels.
I'm pretty sure the baseball bat shows up as a reward from the story mission if you have a skill point in two-handed. It is not a drop in the lighthouse.