I think you might be having a problem with the IR reciever.
I have just made a connector like this myself (well... final build is not quite finished), and what i can tell you is:
the specific IR phototransistor/diode matters a lot.
I have a bunch of 5 and 3mm diodes from aliexpress that just says "940nm" and they have IR filter.
stuff them into the back of the BM257s, and yeah, they pick up, the signal isnt too perfect, but the it semi sorta works, but its EXTREMELY flimsy as far as the "correct" signal being picked up, move the diode half a millimeter and everything goes bust.
What I have found to work, is using a 5mm phototransistor, I should note though, that the 5mm phototransistors dont seem to turn fully on with the signal, whereas the 3mm does. it does however give me around 500mV output. This is not enough to provide TTL high, so i have abused an lm358 as comparator with a voltage divider set to 226mV.
the output of the IR transmitter in the bm257s is as you have guessed very conveniently uart that you can put directly to an ftdi/similar or input on arduino.
I will find up some photos and post today
edit/update:
i looked at your hex dump, i doubt you're getting a correct signal here. the bm257s sends a reading as i recall every 200ms, and the first byte is always 02. What i recommend is that you set the meter in ohms mode and short the leads, that way you should always be getting the same series of 15 bytes repeatedly. if you dont, something is definitely wrong, if you do, it can still be wrong if the photodiode/transistor you use is a bit funky or angled wrong or such.
also btw, a huge thanks to eevblog member "jadew" for helping me with this