Since you wrote that "I actually removed the laser grips a while ago because they were off more than the factory sights," I still wonder whether you really are using the gun properly.
I have to assume that the laser in the grips hadn't been sighted-in by anyone. Certainly, you mustn't assume that the laser came already sighted-in. So if you had expectations that the laser would just automatically point to the correct place, you were making a very bad assumption based upon inexperience.
Usually, however, the iron sights on a factory-made pistol are pretty close to where they are supposed to be. Maybe not perfect, but close enough. So if your Bersa's sights were installed by the factory, they would be the correct height for the cartridges you would be using, even if they might not be finely adjusted for windage (left or right). The factory-installed sights should not make you shoot low.
Therefore, I still have to suspect your trigger technique. You may have everything locked up, and you may not be milking the pistol's grip, but I believe that you are not pressing the trigger while maintaining a proper sight picture. You probably are not following-through, either.
(Besides, if you were milking the grip, your shots would go to the left and down, not just down-assuming that you're right-handed, that is.)
Try This:
Load up. Point the pistol downrange. Put the safety to "off." Now, very carefully cock the gun's hammer while keeping its muzzle pointing downrange.
Now line your sights up on the target, keep them there, and place your finger on the pistol's now-single-action trigger.
Maintaining your correct sight picture, press (don't "pull") the trigger with a slow, steady increase in pressure, until the pistol fires and completely surprises you. Hold everything in place for a count of three. Don't be in a hurry to let the pistol drop down. Don't look for the hole in the target.
Do it again. And again. Fire five shots, each one a complete surprise, and each one with a carefully-maintained sight picture.
Your result should be a medium-size group of holes (not one ragged hole, by any means) centered around the point at which you've been aiming.
Note:
If you're using a bullseye target (a big black circle), don't aim at its center.
Aim instead at the very bottom edge of the circle (on the presumed vertical centerline, of course).
The reason for doing this is that the center of the bullseye is not a distinct point, so maintaining your sight picture is close to impossible.
The bottom edge of the black circle is distinct, and it presents a repeatable aiming point.
Another Thought:
I hate to ask this, but I gotta...
You do know what your sights should look like, when you're aiming. Right?
(My wife didn't, and was shooting high every time, because she absent-mindedly had forgotten how to line sights up.)
The front sight should be centered (left-to-right) in the rear sight's notch, and the top of the front sight should be exactly even with the top of the rear sight. Not above. Not below.
Let me know what happens.