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.
Step 2: Clean up the board
Then I used solder wick to remove all the excess solder from the board. The trick to using solder wick is to put a little solder on the tip of the iron first, so that it can soak into the wick and heat it up quickly. Then place the wick on the pads, holding it with needle-nose pliers, and put the wet soldering iron on top. Then I tend to pull the solder wick so it slides along the board, with the iron sliding along with it, and it wicks up the solder, leaving clean pads. (Slide lengthwise along the pads, and don't push hard, or it can pull the pads off the board.) It can only absorb so much, though, so you need to keep cutting off the soaked part and exposing a fresh end.
The main pad in the middle sucks up heat much better, though (that's kinda the point), and the wick will tend to cool and get stuck, so do the main pad and the pin pads separately, at different temperatures.
It can be hard to see the structure of small shiny things (this IC is only 7 mm wide), so wait for it to cool, clean it with alcohol, and run your finger across it to feel for any bumps or leftovers. In this case, touch is better than sight (just like washing dishes!)
Now that it's clean, you can see all the vias that go through the center pad. You can also see some faint cut marks on the pads from cutting the pins off.
Step 3: Place the IC
I generally just hold the IC in place with my finger, but you might want to use masking tape or something if you don't trust yourself not to slip. ;)
Step 4: Solder the PowerPad
While the pad is not yet soldered, you should still be able to lift the opposite corner of the IC off the board. When you can't do this anymore, you know it's being held down by solder on the pad, even though you can't see it.
Turn up the heat again, and hold the soldering iron to the pad on the other side of the PCB, adding solder and letting it wick through the vias.
This was my first time doing this, and I didn't watch the IC as I was doing it. Since the IC was free to separate from the board, it did, because more solder than I expected pooled underneath the IC and lifted it.
I was initially tempted to just melt it again and push the IC down flat, before I realized how stupid this would be. It would not push the excess solder through the vias! It would just squeeze out the sides of the IC (like squashing a peanut butter sandwich) and there would be solder bridges to every pin. Don't do this!
A better method would be to:
1. Watch the other side of the PCB and make sure the IC isn't lifting off the board.
2. Add solder in very small amounts, let it cool, and then test whether the IC is stuck down or not.
You might also be able to do this by tacking down two opposite corners, so the IC can't lift, and trusting the solder to only wick enough to fill the pad, and not to squeeze out and touch the pins. This would probably work even better, but I didn't try it, and you'd have to desolder one corner in order to make sure it couldn't lift anymore.
After this mistake, I used solder wick to suck the solder back out through the vias, until the chip lay flat again. Phew! Done with the tricky part.
Step 5: Solder the pins
In my case, two of the pins got splayed out while wicking away solder, and I had to carefully bend them back into position with tweezers. I was very afraid of breaking them off and having to start over again. Luckily, they were redundant, so I could have broken one off and survived.
Step 6: Clean the board and check everything
After visually confirming there are no solder bridges, use a multimeter to check every adjacent pin for shorts (usually by touching the components they are connected to, not the pins themselves), and then check every pin for shorts to the power pad (obviously it's ok if a grounded pin is connected to ground).
After you've confirmed that there are no shorts between any pins, you're done! Plug it in and try it.