I was figuring you could weigh it carefully, measure its volume as accurately as possible (dunk in water, measure the overflow), and calculate its density. If it's near solid steel's density, it's probably AP.
But then I figured, with the comparatively small weight of explosive it might have, if it's HE it might not show up as much of a difference in total density. (I think with the 16" naval rifles, the HE shell of around a ton only had something like 200-250 lb of explosive in it.)
So I figured I'd write this out in case somebody else got this bright idea which ain't so bright.
Have them try to detonate it. If it goes, you wouldn't have wanted it around. If it doesn't, it will probably stay in one piece (if you can find it again) and you can probably keep it as a deactivated souvenir unless you live in "certain States" and probably some municipalities.
Good advice to not bring it in, but park it somewhere "safe" and call them and their equipment to it.
If I recall correctly, small arms HE ammunition was stoked with tetryl, which needed no detonation or fusing mechanism. The tetryl just went off by itself on impact.
Terry