The problem is probably insoluble in Special Relativity alone*, because of the available clock-synchronization schemes in a universe populated only by the self-contained experimental apparatus (sender + local clock, receiver + local clock), experimental pulse of light, and flashing-light morse code or whatever to compare timestamps. If we are allowed to add more matter of our choosing, we can probably measure the one-way delay with good precision.

The absence of gravitational sources is an aspect of the flat spacetime of Special Relativity, but if we are cheating by adding in (and declaring gravitationally negligible) the experimental apparatus, why can't we cheat by adding in a non-gravitationally evolving bit of matter which can serve as a clock? A low-mass, sparse, spherical, uniform cloud of of hot dust expanding adiabatically can serve as a clock by measuring its and temperature if the one-way-transmitter and one-way-receiver are freely falling within it and moving slowly compared to light. This is essentially a demotion of the sparse cosmic microwave background gas/dust of massless photons -> sparse gas/dust of neutral low-mass molecules. The CMB expands and cools, while we're within it. Our non-relativistic molecular gas expands and cools, while our one-way test equipment is within it.

Of course, what is too much of a cheat in Special Relativity and what is not is debatable. In all the cases above we are ignoring the Raychaudhuri equation with the only justifications being that the timescales are too long to tell if we're focusing, and we aren't obviously engaging post-Newtonian (PN) corrections. (What gets us into trouble with PN formalisms in GR can get us into trouble in gravitation-free SR though: ultraboost one side of the experiment, rather than "... moving the clock slowly with respect to the stationary one". You guessed correctly that boosts and accelerations could be a problem in (2)&(3). However, contra your (3) acceleration is perfectly permissible in "pure" SR and the result is only equivalent to being in a uniform gravitational field (rather than with a potential gradient), and only somewhat briefly (you can rest your clock on an enormous rocky planet for much longer than you can accelerate your clock at ~ 10 g). The time dilation in (3) is Minkowski / Born / von Laue / Einstein 1905-1911 Special Relativistic and not post-1915 General Relativitistic. Your (1) is done for you for free in "pure" SR, since there is no gravitation there.)

General relativity is hardly a panacea: if we have a strongly expanding vacuum our one way pulse might never reach the detector. In a dynamical curved spacetime we can break the symmetry between legs of a reflection 2-way test in any number of ways.

It's really the breaking of the vacuum condition that lets us set up a "global" or at least wide-enough-area clock. When we're allowed to introduce half-life decays or thinning background matter or radiation, or distant millisecond pulsars, we are more likely to be able to use a synchronization scheme sufficiently different from Einstein's method and successfully compare timestamps at the sender and receiver of a one-way flash.

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* If we assume Special Relativity then we already have global Poincaré invariance, so we have already have symmetrical legs of a reflection test. If we have a setting which is maybe Minkowski spacetime, or maybe something other than Minkowski space that breaks the symmetry of the legs in a reflection test, then we probably can't do it with a one-way test along the lines you're thinking.