Decrypting LFSR, knowing the beginning of the cyphertext, and nothing else
It's used the right tools, apparently with the right input, but missed how to use their output.
If, when given the first 84 bits of keystream, a proper Berlekamp-Massey implementation outputs a polynomial of degree 27, then it also has found a 27-bit initial state, such that the 27-bit LFSR with this initial state has output starting with the 84-bit keystream. And one can have high confidence ($p\approx1-2^{2\cdot27-84}&gt;99.9999999\%$) this is not by accident.
Next steps:

Understand where in the Berlekamp-Massey code used lies the 27-bit initial state, and output that. Basically, if the LFSR convention is that of a Fibonnaci LFSR, that initial state is the 27 first terms of the input sequence analyzed (which is why it is often not an explicit output).
Make an auxiliary program that implements this LFSR and produces keystream.
Verify correctness of the above by checking that the first 84-bit output is the input to the Berlekamp-Massey code / the known keystream. If not, fix whatever is wrong.
Generate as much keystream as needed to decipher.

Note: I recall I helped fix Berlekamp-Massey code in C. Try it online!. And someone wrote some elements of the auxiliary program (but it's non-trivial to link the two). Help on use of a particular tool is off-topic, especially for CTF or homework. And writing the auxiliary program (perhaps as an extension of the  Berlekamp-Massey code) is a simple and essential exercise.