So if you had a Turing machine traveling along this wordline, it could send a signal to p if and onliy if it halted, and the observer at p is guaranteed to receive the signal if the machine ever halts. No infinite-precision measurements are involved (unless perhaps you believe that a Turing machine operating reliably for an indefinite period of time is tantamount to an infinite-precision measurement).
Even if such spacetimes were possible, in order to exploit them for hypercomptation you would require a true Turing machine with infinite tape, infinite energy supply (unless it was perfectly reversible) and enough durability to run for a literally infinite amount of proper time without breaking. Such requirements seem inconsistent with the known laws of physics.
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You can't compute a prefix of Chaitin's omega of any arbitrary length. You can compute prefixes only up to some finite length, and this length is itself uncomputable.