Yes, victor being helpful as always...
We got the supervisor password! In the end it was "NCD"
It felt so good to finally boot up the computer & reset the password

Now the HDD password is gone, too!
At the end of the day, the successful read I made was done using an Arduino Mega 2560 (pictured below) but any microcontroller should work.

I used this github code to read the chip:
https://github.com/0xJoey/Arduino_93C46/tree/master
The "read" sketch in the example code got me started, but at first it was giving me garbage data. By cross-referencing the 93C46 spec sheet I learned this code had a
critical flaw; it was sending 8 bits of the address in the read command (for 64x16 mode), but it should only be sending 6. One line of code was all I had to change to get it working.
(line 139 of 93C46.cpp, for reference; change to "send_bits(addr, 6)")
Once I made this change, I attached my clip to the EEPROM chip (de-soldered from the mainboard) & plugged the arduino into my computer. I fired up GNU screen on the serial console and took a screenshot of the output given by the arduino; then I used hexedit to manually key in the bytes & save the dump. (Next time I do this I need to find a way to read the bytes directly from the serial device & output those to a file... I did it the hard way this time!)
OP, if you have an Arduino lying around (or something compatible with Arduino IDE like a Pi Pico or a Teensy), then let us know if you give this a try!