No, "meaninglessly determined" vs. "meaninglessly random" is a false dichotomy, and it only arises under the assumption of a wholly mechanistic universe. (Yes, even our conception of absolute randomness, especially as as being meaningless, is framed by our unconsciously mechanistic outlook.) Free will is not meaninglessly random, though it is unpredictable and not deterministic.
Another thing to consider is that when you pose two possibilities as the only logical possibilities and allege that in both cases free will is logically ruled out, you're essentially ruling out free will by definition; and if free will were really ruled out by definition then there would be no need to bring up those two possibility-branches because they'd be irrelevant. So obviously you're duping yourself somewhere.