You caught it pretty early thankfully. It puked HARD though.

Instead of desoldering the battery holder, snip that thing out. Hold it upside down over a trash can and just let it drop in. Neutralize with vinegar. Clean with Alcohol or Naphtha. Some wire brushing. And then see where you are.

Don't forget to inspect the connectors below too.

Use remote battery holder, or better yet a FRAM chip.

The pads lift from mechanical stress. Heat just makes it weaker.

Pads flake off when you rub the tip against the pcb. Through holes and other pads get ripped out when you pull a lead through they hasn't been desoldered all the way and you try and pull the through hole out.

If something doesn't desolder well. Add more fresh solder and try again. The hole / pad must be filled in with solder to desolder it. A halfway cleared out hole with not desolder properly until you fill it back in. Surface tension and heat transfer.