This drives for maximal use of scrap, cheap but labor intensive. If you use a core, perhaps the combined cores from 10 to 30 discarded microwave ovens could help.
How is the strength of the magnet scored? It is really driven by ampere turns. How long does the field have to last?
Based on what i have seen so far, my approach would be on the order of
10 turns of 1/4 inch copper tubing supplied by 1 or 2 lead-acid (auto) batteries.