So, I did this a while back for reasons that I don't exactly remember at the moment... but while I don't recall the reasons, the process I came up with worked well enough that I made about 300 LPP .38 special cases. I later turned them all into .380 rook cases, but that's probably another story.
I got a 3/8 collet for the lathe that opened just enough to accept a sized case. I used a collet stop so that the brass would slide in and lock down consistently at the same depth +/- .001". I used a 5.3mm Carbide end mill to machine the PP to depth of .120 using a micrometer stop. Then they went into the primer pocket swager, followed by deburring the flash-hole. Primers seated fine, I reduced loads slightly, and other than slightly wider SD's all was well with the world.
Getting everything set up took some work, but once it was all done, the conversion was easy, and it didn't take too long to do 100 pieces of brass.