So in your case the BPM runs as the app account running the Epicor App Server instance so writing to a network drive is gonna be tricky.
It should work, however you are correct Async BPMs are broken to hell and back in all versions of e10.
They mostly ish sometimes work, but they fail silently and accumulate for ever. I have it on good authority this is being addressed we’ve reported these bugs many times, but as of right now, async is a no go. IMO
However I’m guessing your issue is a permissions one, try writing to a local file in th C:\ drive of the Epicor server.
I had one running for a month and it made something like 100K entries in the AsycBPM tables because 1 of them failed. As soon as ANYTHING fails everything else just stops processing.