Can I use a differential that can be traced through the whole cipher with 100% probability?
Usually differential cryptanalysis relies on something called the "wrong key randomization hypothesis", which is the assumption that decryption of the last round with the wrong key results in a random difference at the beginning of the last round, while decryption with the correct key will result in the expected difference with the probability of the differential (which hopefully is sufficiently high or low as to be distinguishable from a random difference).  However, if there is a probability one differential going through the entire cipher then by definition that means that decryption with any key will lead to the same difference.

You might try linear cryptanalysis instead.  A probability 1 linear characteristic through the entire cipher will give you 1 bit of the key with just a single known plaintext (Edit to add: poncho adds a good clarification - the probability 1 linear characteristic must involve at least one key bit to be useful for Matsui's algorithm one).  If for some reason there is no linear characteristic through the entire cipher, then you could try differential-linear cryptanalysis using that probability 1 differential through most of the cipher and a short linear characteristic at the end.