Concur with the Buffalo Bore hard cast flat-nosed lead. It's not going to expand but it will penetrate just fine and the flat nose will be more effective than FMJ.
It shoots and feeds without problems in my Bersa Thunder 380, and you can really tell the difference in recoil with their "+P" labeled ammo.
Keep in mind that there is no such thing as .380 +P so what you are getting is something loaded hotter than SAAMI specs. You might instead consider their standard pressure load in the same bullet type.
As far as using what you have on hand- if the JHP plugs up with cloth, then it will function like an FMJ anyway, so there's no real reason to change. The JHP isn't going to bounce off a winter coat. It just may not expand. Or, go with the FMJ, your choice really.