I've read accounts of people replacing the burr sets on their Lelit PL53s, but what method did you use to hold the burr shaft stationary while loosening the reverse threaded nut above the burr?

There's a hole on the bottom of the lower burr that you could (theoretically) use to hold the whole thing steady. I didn't realize it was there when I tried to undo mine. I ended up taking the base apart and chewed up the bottom of the shaft where it connects to the gear/motor by clamping a wrench to it. So, don't do that.

I see the hole you're referring to; unfortunately, that hole on the bottom of the burr falls in perfect alignment with one of the 3 plastic sweeper arms. A few days ago, I too disassembled the upper portion of the grinder down to the gear reduction drive to put a vice grip on the burr shaft. I decided against for the reasons you mentioned...hmm. I'm afraid using a screwdriver to jam the plastic sweeper arms would break them as the torque on the nut seems relatively high.

Not sure if it will work for you, but here's what I do.
Leave the upper burr carrier and the wormscrew adjustment in place, fill the grind chamber with rice, turn the nut as if you're using a hand grinder. The resistance of the rice should be enough to hold the burrs while you turn the nut.