How could I safely fix my walking "dead pixel" bug?

Leave it alone. It'll eventually die and find its way to the bottom of the LCD.

I've had the exact same problem for around a week, and while this question may be amusing, it could end in disaster if you squish a bug and need to dismantle expensive equipment.

What I noticed was after a couple of days, it appeared to be dead, but I lightly tapped around it and it didn't move.

The key is, if it doesn't move when you tap around it, it's still alive because it still has a grip.

A couple more days passed, and I noticed it was stationary again. With one light flick around an inch beside it, the bug fell right to the bottom of the display, out of sight and out of mind.

Happy screen-flicking!

This thunder bug issue happened to me 2 times. The 1st time I was stupid enough to squash the bug inside the monitor ended up sending the monitor for repair. It was an expensive lesson. The 2nd time, which just happened last night, I quickly turned the monitor light off and left a bug zapper light (which attracts bugs and electrocute them) overnight. I've heard a big zapping sound middle of the night and the next morning, the sucker has completely gone :). The trick is to keep the monitor dark and attract the bug with another light source. You can get a bug zapper around 20 Euro. Definitely worth it to protect your expensive monitor.

Check for an Apple Store nearby. In or out of warranty, they may be willing to remove it for free due to the novelty.

If you don't mind cracking the case open, grab a nice set of small tools and head over to iFixit and find the Teardown guide to your model. Here's the 13" Macbook Pro unibody LCD teardown. Pluck the guy out and blow out some dust while you're there, and put it back together.