As long as they are loaded to spec and pass the plunk test on your individual handgun, there should be no issue at all. The Bersa 380 CC Thunder that I had seemed to eat anything - I got rid of it but am now thinking of getting another one.
All I shoot in my reloads are cast bullets that I cast up. If the cast bullet is sized for the size of the bore, there should be no "leading" issues. I don't have a .380 at the present time but I do reload 9mm for a SR9 and a 9mm Shield - they work great and no issues at all.
Some semi-autos can be a little "finicky" when it comes to the bullet design - i.e. a round nose may feed better than a semi wad cutter or similar design or vice versa and some like 'em all. Like any reloading, you need to work up your load for your particular handgun as such things will dictate how your particular handgun cycles. As an example, my 9mm Shield has a rather stiff recoil spring as opposed to my somewhat larger SR9. A 3.5 grain load of Bulls Eye under a 121 gr. round nose lead bullet will cycle the SR9 . . but not so on my 9mm Shield which seems to work and cycle much better with a 3.7 to 3.9 grain load of Bulls Eye under the same bullet. Not a problem though as I load for the 9mm Shield and the slightly heavier charge that shoots and cycles the Shield with good accuracy works well in the SR9.
If you are going to reload for your .380 Bersa, just make sure you use accepted loading data for the powder and bullet you are using whether the bullet be lead, copper clad, etc. I mention the plunk test which is loading a cartridge and then with the barrel removed from the pistol, dropping the cartridge into the barrel chamber - if loaded correctly, it should "plunk" and seat in the chamber with the correct headspacing - since it headspace on the throat of the casing. A handy tool to have is a "cartridge gauge". Once you loaded cartridge comes off the press, you drop it in the gauge and if the base is even with the top of the gauge, your cartridge is within specs. Once you get your dies adjusted and set and the loaded cartridges fit properly in the gauge, you can check them randomly the same as you wold your powder charges that you are dropping.
Good luck to you - the Bersa is a nice hand gun often "under appreciated".