I succeeded in reprogramming the eeprom in my UT210E after minor struggles. Just reporting here how I did it, maybe it helps someone. Most (all?) of this has been mentioned here and there in this thread before.
Got the TL866A programmer with a lot of adapters from an Amazon seller, downloaded the software from xgecu.com. I hooked it up to the eeprom using the great little probes that came with my Saleae 8 bit logic analyzer. These probes are small enough to easily hook the legs of that eeprom without crowding out each other. They also have two pin connectors each, making it easy to hook up both the programmer and a scope probe to the same gripper.
Initially, I couldn't get the TL866A to program the eeprom, though it could read it out just fine. As you can see in the scope image, where CH1 is SDA, CH2 is SCL, and CH3 is WP, WP goes high during programming, which isn't right. (I'm also not clear on why SCL is missing during a large section of the programming.) I don't know if the programming isn't driving WP right, or if the UT210E pulls up too hard, but simply connecting WP directly to ground fixed that, and programming could proceed without a hitch. I should add: don't forget to set the main rotary switch on the UT210E to anything except "off". In short, I didn't need to modify the UT210E in any way, unsoldering anything, or adding in connector pins.
The original contents of the eeprom in my UT210E isn't exactly the same as others I've seen in this thread, so I'm including it here. I'm also including the changed contents after programming. My UT210E has the later main chip, the 1106.