IIRC, I've seen forum or Facebook chatter that LAPD allows FMJ for 380s, but I haven't seen it in writing. I can say for a fact that .380 is the one caliber where Chicago PD allows FMJ..380s can be finicky. My Glock M42 took several hundred rounds (don't recall now, but I wrote it up here) before it would work right with carry ammo. I'd plan on running at least 300, preferably 500 rounds of FMJ before trying duty/carry ammo.
Some pistols don't work right with some ammo; probably tolerance stacking. You may need to try other ammo. I don't recall what it is or where I saw it, but LAPD only allows one load for .380s; I'd try to find out what that it is and try it.