G'day mate, I just happen to work with stuff like that where you get broken bits and pieces, especially decapitated stainless screws and bolts, but since you have a aluminium base to work around it makes it more difficult. Allas do not despair, for the piece you want out is led covered with copper. What I would do is this:
Fix the press securely where it won't move on you, then get a sharp punch marker. Locate the middle of the bullet and punch mark it. (the flat
side, or the back of the bullet)
Then get a small diameter e.g. 1-2mm long shank drill bit (sorry I,m not good with imperial measures) drill the marked point about 3/4 of the lenght of the bullet. Increase the bullet diameter and drill it out again,keep increasing the drill bit and repeat this until it is safe, (meaning, do not hurt the press ). You have two options here ;
1 As someone already mentioned to get a screw drive it in and gently wriggle it out.
2 This is what I would do (you will need a steady hand for this though) get a 6 or 8mm chisel for wood, and gently cut down where the cavity is between the 2 dies and the remaining bullet diagonally down as far as you can. if you do everything right the bullet will be cut in half and should collapse after cutting. Then use a long nose plier to dislodge the remaining bullet just like a dentist would, when he is pulling out your wisdom tooth. (at this stage some tapping might help from the opposite direction with an appropriate size wooden dowel).
Regards and good luck to you and to everyone on the forum villihun