I did what your proposing and it worked out well (see this post) If you have no magnetic sense, it’s definitely because the magnet is dissolving inside your finger. The only alternative explanation is that it somehow reached the Curie temp for a few minutes, which is 150°C so probably not. The puffiness is another sure sign.
I would get it out as soon as you can secure another magnet (or before). There will probably be a large pocket of scar tissue around the magnet too, like I had. Make sure that’s removed too, because that’s where most of the metals have been captured. To get a new magnet, some body modders have them, but they’re not great. Parylene or silicone. Just be careful