IDK about the LCI/II/III as I never got the card I have working on my LCII beyond maybe installing the drivers and a green light of some sort.
But I will note that I had similar trouble with my Power Mac 6100/66 not playing nice with a more modern 10/100 Mbps (Fast Ethernet) switch or hub, whereas interposing a 10 Mbps hub in between it and more modern hardware solved that problem (probably a similar auto-negotiation problem).
I wouldn't necessary rule out sudden failure of some critical PSU component, these machines are 25 years old after all.
A multimeter is okay for simple voltage, resistance, and continuity checks, but not generally useful for determining if capacitors are dead (at least not while they're in-circuit) or the overall health. IDK if there are any good test points, but it wouldn't be a bad idea to verify that the inputs/outputs of the regulators seem as expected and that where the electricity enters the board the voltage also seem correct. Also capacitors don't have to look bad or be leaking to be dead/out of spec, or at least that's my understanding.
Does it matter whether you leave the battery out or not?
I really doubt it matters here, but compressed air is pretty cold coming out.
I will try to remove the anode cap by hand once I can cobble together a tool to discharge the CRT. An flathead screwdriver (with an insulated handle) and an alligator clip running from the screwdriver to the metal chassis should be sufficient to discharge the CRT, correct? (That's what I've seen in almost every video showing someone discharging a CRT.) I'm assuming it's crucial to remove the logic board from the computer before doing any work on the CRT.
As mentioned, I'm mainly worried about the neck board and reinstalling everything. Last thing I want to do is either break the neck of the CRT or reinstall something wrong and let the magic smoke out of something else or, worse, catch something on fire.
As for now, I've made a video depicting the behavior this machine is currently going through. I was able to get it to boot into the ROM operating system, but it instantly crashed as soon as I loaded a floppy disk. When it came back up, it went into a "chime cycle" where the machine would crash (thus chiming again) when trying to eject the floppy. I ended up having to manually remove the floppy using a straightened paper clip.
Here's the video: