What a coincidence, I took one of my TPP0250's apart a couple of days ago!
The probes appear to be compensated inside the scope, not on the probe itself. The scope keeps track of the probes it knows about using their serial numbers, and applies compensation appropriately once each probe has been calibrated on each channel.
After figuring out what was inside it, I ordered up some parts and last night I spent some time looking at the 1-wire. The scope simply reads the first 112 bytes from the EEPROM in a single command. I have not seen any other activity. The scope treats each readout channel as the only device on the bus, it does not read the 1-wire device's serial number. I didn't see any writing back to the probe.
Comparing four probes, the last byte looks like a CRC or checksum of some sort, everything else is identical apart form the serial number of the probe, in clear ASCII, as is the probe model number. It doesn't look like a simple checksum, because the serial numbers of my probes are very close, all within a dozen or so serial number-wise.
I managed to copy the EEPROM into another blank DS2431, and it was recognised as a clone of the original probe. One caveat, you need to load the bus with a 3.9k resistor as well as the 1-wire device, it won't recognise or attempt to "probe the probe" otherwise, the scope wants to see 3.8V on the readout pin.
I tried doing 1 bit changes to the serial number, but the scope won't then recognise it as a valid probe, so the checksum/CRC is definitely used.
So I spent some time with a CRC8 check, seeing if I could massage the polynomial and start condition to come up with a consistent result for all four probes by an exhaustive check, but no luck I'm afraid. I would say, though, that I don't have experience of this kind of entertainment, so I am sure there are other methods. It may well not be a CRC at all of course.
Probe name is at offset 0x0c ("TPP0250\0"), and serial number is at offset 0x14 immediately afterwards ("C######")
05 03 31 70 9E 6B 04 03 10 5D 14 02 54 50 50 30
32 35 30 00 43 30 33 36 31 34 32 00 00 01 06 12 <- C036142
4B 1B B6 31 06 16 80 96 18 4B 06 18 14 38 89 2C
06 14 28 6B 6E 4D 06 1A 00 00 96 43 06 10 CD CC
CC 3D 0E 5D 00 24 74 48 00 1B 37 49 00 7C 12 48
17 8A 01 E0 9E 14 4B CC B2 64 2C 7C 6F FC 72 77
6F 3E 2E 00 00 C8 44 08 85 01 00 02 03 01 04 3D <- 0x3D checksum/CRC
05 03 31 70 9E 6B 04 03 10 5D 14 02 54 50 50 30
32 35 30 00 43 30 33 36 31 35 33 00 00 01 06 12 <- C036153
4B 1B B6 31 06 16 80 96 18 4B 06 18 14 38 89 2C
06 14 28 6B 6E 4D 06 1A 00 00 96 43 06 10 CD CC
CC 3D 0E 5D 00 24 74 48 00 1B 37 49 00 7C 12 48
17 8A 01 E0 9E 14 4B CC B2 64 2C 7C 6F FC 72 77
6F 3E 2E 00 00 C8 44 08 85 01 00 02 03 01 04 59 <- 0x59 checksum/CRC
05 03 31 70 9E 6B 04 03 10 5D 14 02 54 50 50 30
32 35 30 00 43 30 33 36 31 34 30 00 00 01 06 12 <- C036140
4B 1B B6 31 06 16 80 96 18 4B 06 18 14 38 89 2C
06 14 28 6B 6E 4D 06 1A 00 00 96 43 06 10 CD CC
CC 3D 0E 5D 00 24 74 48 00 1B 37 49 00 7C 12 48
17 8A 01 E0 9E 14 4B CC B2 64 2C 7C 6F FC 72 77
6F 3E 2E 00 00 C8 44 08 85 01 00 02 03 01 04 D1 <- 0xD1 checksum/CRC
05 03 31 70 9E 6B 04 03 10 5D 14 02 54 50 50 30
32 35 30 00 43 30 33 36 31 35 34 00 00 01 06 12 <- C036154
4B 1B B6 31 06 16 80 96 18 4B 06 18 14 38 89 2C
06 14 28 6B 6E 4D 06 1A 00 00 96 43 06 10 CD CC
CC 3D 0E 5D 00 24 74 48 00 1B 37 49 00 7C 12 48
17 8A 01 E0 9E 14 4B CC B2 64 2C 7C 6F FC 72 77
6F 3E 2E 00 00 C8 44 08 85 01 00 02 03 01 04 CE <- 0xCE checksum/CRC