Go for it.
I've been handloading for 2 BG 380s (My dad and I both own one) for quite a while now, and have not noticed any bulge.
My dad had a Glock for a few months so I'm intimately familiar with the bulge (if it's got a rectangular firing pin impression it's probably bulged).
While working up a variety of loads, I went well into the high side of the pressure range several times, and though the primers were flattening, with a bit of crater showing, No bulge was noticed.
The cases all sized fine through Lee Carbides I use. I do give the cases a spray withg diluted Lee resize lube because it makes the process require much less effort. I like to use a hand press to resize pistol cases. The .380 operating pressure is quite a bit lower than the 9mm and I'd guess that's why it shows up in the 9mm but not the 380.
Use a Lee factory crimp die to finish, and you'll be sure the ammo will chamber. It has a carbide resizing ring to ensure that the cartridge maximum dia is under SAMMI specs.
I've loaded brass up to 5 times with this process, have had no splits, and lengths are still consistant and under spec. Time wil tell as they go through more cycles. We have about 1000cases in rotation at this point.
I know I sound like a Lee salesman, but it's what I use for most pistol calibers. I do weigh the powder on a RCBS 1010 and use very old HORNADY measure in the process.