Hi Bronco,
Given that the electromagnet route seems problematic to say the least, I'd like to suggest an alternative. Take a physically strong, non-magnetic container like an aluminium or plastic container (bucket, saucepan with side handles etc). Insert a strong permanent magnet into the container with its poles facing out through the base, and epoxy it into position. Tie a strong, eg nylon rope to the handle of the bucket (or handles of the saucepan) and lower that down the well. Let the rope take the strain. The same concept would also extend to the electromagnet, if you want to continue down that route.
For this, you would place the electromagnet AND battery into the bucket/saucepan. You would use a relay to switch current from the battery into the electromagnet, with a light-duty cable connected to the relay. this light-duty & low weight cable would be taped on to the rope. You would connect the free end of this cable to another battery or wall wart to energise the relay.
If the small standoff due the thickness of the base of the container causes unacceptable losses in magnetic lifting strength, consider cutting suitable holes into the base, to allow only the poles to protrude outwards.
You mentioned using a winch. If so, then you could replace the rope with a rope lifting harness. the harness would use the same knots tied to the same handle(s) as in the previous case, but would only extend a short distance above the container, ending in a loop. The winch hook would engage the loop, with suitable safeguards to prevent the hook slipping out inadvertently.
MrAl made a good point about non-magnetic scrap like lead. If that is a strong possibility, consider hiring a crane with mechanical grabber facilities, like this one in a scrapyard:
https://www.shutterstock.com/pic-37273147/stock-photo-mechanical-grabber-working-in-a-scrapyard.html