Wow, thanks for all of the responses. Extremely helpful. I'm very new at reverse engineering (if that's what you want to call this) and I'm absolutely a blank slate when it comes to recognizing different chips.
Anyway, I'll do my best to respond/answer what you all asked/stated.
John, it seems like others have identified the black chip to the left of the blob as EEPROM. That makes me wonder... where is the conditioning chip at? It has to have one to go from analog to digital, doesn't it?
I'm also trying to avoid messing with the raw analog signal because I'd like to maintain the accuracy of the scale...I feel like if I messed with the analog signal, I would have to convert it "by hand" to get an actual number, which would be prone to inaccuracies... although if I'm wrong, please correct me (seriously, I'm trying to learn. If I say something stupid, call me out).
FM, I realize this is an incredible noob thing to say, but how do I probe them? Do I need a several hundred dollar oscilloscope (seen that mentioned on some blogs)? Regardless, how do I probe something that's so small? I'm very unfamiliar with traces, as the only electrical education I have is with breadboards and larger circuits. If you could send me a link or just some incite on how to get started with that, I'd love to probe everything on there even if I solve my issue before then.
Also, this is the only chip. The red and black wires go to batteries (as expected) and the blue/white wires go to the actual scale. That must be where it gets all of it's raw data from.
Tom, FM had suggested that perhaps T5 is GND. After looking at the chip datasheet that you found (clearly the correct chip, I didn't think to look up the serial number thing), it's obvious that T5 is SCL and T* is SDA, however, how did you know that T2 is GND when the EEPROM chip doesn't actual have a trace to it? I noticed that there's a faint trace going around the entire chip that touches T2. Is that how you knew?
Here's the most obvious question that will absolutely solve my problem, but how do I connect those 16 display pins to an arduino pin? I assume I have to solder things at this point, which I'm excited about (just bought a kit) but how do I actually go about doing that? Do I intercept the signal before it hits the ribbon (To the right of the black blob)? After that I'm very familiar with testing for the sequence, and I should be able to handle that.
Thank you all so much for the responses, you were incredibly helpful. I feel 100x more confident now, and if any of you have book recommendations/websites for the extreme amateurs that can help get me started with this hobby, please feel free to post them.