Does accumulator == stack?? If not, how should anything ever be added to the stack? The lack of explanation (for everything) is a big problem.

Your last sample test also seems to be broken. The s is never pushed to output.

most insignificant digits

It's "least significant bits".

RGB888 = 4c6a02
R = 4c (hex) = 01001100 (bin)
G = 6a (hex) = 01101010 (bin)
B = 02 (hex) = 00000010 (bin)
BGR555 = 000000110101001 (bin) = 425 (dec) = 01a9 (hex)

Expected: "001a"
But was:  "a901"

Either the reference solution is wrong, or some information is missing from the description.

Edit: I've looked at your solution, and it is indeed incorrect.

Remove the three most insignificant digits from each byte to get a 5-digit binary number (ex. 10001).

What if it's already less than 5 digits? What if it's 6 digits long?