Dying of embarrassment, I decided to try the impossible and replace it before anyone found out what happened. I've soldered surface-mount ICs before, but never any with a PowerPad on the bottom. These are especially tricky to do by hand, since you need to melt the solder under the chip, without making any solder bridges between the pins and the pad. I wasn't sure it was even possible to hand-solder.
(The reason I was able to do this is because there are vias connecting the PowerPad to the other side of the PCB, so that the ground plane on the other side acts as a heatsink. If your design doesn't have these vias, or the holes in the vias are too small for solder to travel through, this method won't work.)
But I was successful! Now no one needs to know my secret shame.
Step 1: Remove the old chip
To remove the busted chip, I first cut all the pins off of it. That way I didn't have to desolder both the pad and the pins at the same time; I could just focus on the pad. I used an exacto knife and pressed it carefully against the pins, one at a time, as close to the chip as possible, until they were all broken off. I ended up cutting into the PCB a little, as you can see in other images, but it didn't harm the layout.
Since the PowerPad-style chips use the PCB as their heatsink, you're supposed to create a bunch of vias right under the IC. This is the key to removing it. If you don't have these vias, I don't know what to tell you. Get a hot-air rework tool, or try to wick solder under the chip from the sides, I guess.
So then I turned my soldering iron up to a higher temperature than normal and held it to the pad/vias on the other side of the board until the solder melted all the way through. The chip came loose and separated from the PCB, and I was then able to get under it to free it the rest of the way.