Ok... the new MLX90614ESF-BCI is in, and the EEPROM dump is done. Here are the relevant registers (the positions that can be written by the user):
00: 9993 -> ToMax
01: 62E3 -> ToMin
02: 0201 -> PWMCTRL
03: F71C -> Ta Range
04: FFFF -> Emissivity
05: 9F75 -> Config register
0E: BE5A -> SMBus address (LSB only)
0F: 0800 -> Melexis reserved
09: D15A -> ????
Almost all of these registers are the same between the two sensors. The only exception at this point is 0x09 (undocumented), where I think the calibration is stored somehow. Which is relevant because once I fed these new numbers into my older sensor it's sort-of-working again... except now the temperature is just 3~4°C off. I can live with that (the important thing for me is the difference between measured temperatures, not the absolute temperature of any particular point), but correcting it would be nice.
Anyway... if anyone decides to post his/her numbers it would be interesting.
I found why one particular memory position wasn't being written. I checked the new values time and time again, including the CRC, and everything looked fine, so I didn't know what was going on. Well... before anything gets written to any register, that register must be zeroed... and that process ALSO has a CRC, which for one particular register (0x01) was not correct in my code. That made the zero-ing to fail, which in turn prevented the new value from being written. After changing the CRC (and another one on the cycling) everything seems to be working fine.
PD: be VERY careful when messing with register 0x05... from the literature changing bit #3 makes the factory calibration to get out of whack