Can a thrust bearing be changed with the engine in the car? (318)

You can without pulling the engine, but truthfully you're making a lo of work for yourself by avoiding it. Centerlink, mounts, pan, losen the center main, and carefully drive it out from the non-tang side. Tap the new one in from the "tanged" side. Replace the cap, torque it, losen the bolts, and then you need to pry the crank forward and back to aligned the halves, and re-torque it.
Question is, why did it go? You're fixing a symptom. Not the problem.

Don't I have to do everything whether I take the engine out or not? Part of the problem is I don't really have the room to take the engine out, especially if I end up not being able to fix it in one day. I'll be doing the work in someone else's garage and I can't take up the whole space.

As for the question... I don't know, you tell me, what makes a thrust bearing go bad?

I think it's bad because the crankshaft moves forward and backward about a quarter of an inch, it can be moved with a lever and you can see it moving with the engine running. I haven't found any metal in the oil though so I'm hoping I caught it in time before it did major damage to anything else.