Just one note, the style of the head is most probably going to be a standard cap head (allen) hex.
You mentioned that you wanted button head, this style is a very flat slightly raised in the middle and will not fit in the stem, I'm not to sure of the name of the taper head allen screw you are refering to.( if it has one )
However you can very easy reshape a standard cap head allen screw in to the standard shape used on the stem. Simply put the screw thread into a drill and lightly clamp in the chuck and then while rotating the drill you can emery paper the end of the screw head to your desired shape, I have done this many times to produce custom screws. After you have got the shape you can then polish with 1500grit paper and then autosol metal polish to get a mirror finish.
Just had a quick look now and the name comes under a couple of styles either (taperd cap head) or (conical head)