There are 2 issues with flat cables:
Issue 1: Ethernet cables have 8 wires internally, each pair of 2 wires is supposed to be twisted to greatly improve signal integrity.
See this image, top cable is an example of a cable without twisted pairs while the bottom cable has twisted pairs.
Most flat ethernet cables do not have twisted pairs (as it's hard to make a thin cable with twisted pairs).
I bet if you were to cut open your flat cable it would look like the top cable in that image (just with 8 wires instead of 4).
Issue 2: Flat ethernet cables usually have less shielding.
Shielding helps a cable to maintain connection quality over longer distances.
This means that flat ethernet cables generally don't work well over long distances.
I see a few options for you to consider:
Option 1: There are a few (not many) flat ethernet cables that do have twisted pairs, here is one.
Option 2: Ethernet relays. With an ethernet relay you could use two shorter cables (ex. 25ft instead of 50ft) with a device in the middle that acts as a repeater. That would significantly reduce signal degredation over the distance of the run. If you don't have an outlet near the middle of the cable run you can use an ethernet relay (ex. this one) with a POE injector (ex. this one) on one end of the run. If there is an outlet near the middle of the cable you can just use a cheap Gigabit ethernet switch.
Option 3: A different low profile cable. There are compact cables other than just flat cables. Here is a high quality slim round ethernet cable.
Personally if I were you I'd try option 1 to see if it works, then if not I'd try one of the other options.