The way out of this dilemma is simply to recognize that the one way speed of light and the synchronization are two parts of the same convention. It makes no physical difference what convention you pick, so pick an easy (isotropic) convention. Since it is merely convention you are automatically justified in doing so, and you recognize that people who want to make extra work for themselves are also entitled to pick a silly convention.

Hm, is it really pure convention? I doubt it. Using SR as a spacetime model, which includes the assumption of isotropy and homogeneity of space for any inertial observer, of course implies the consistency of the standard Einstein-synchronization convention, but as soon as you take gravitation, and thus GR effects into account, you'll find deviations (like the gravitational deflection of light running close to the sun, as was one of the classical confirmations of GR, making Einstein the first superstar of science when this effect has been observed by the Eddington collaboration in 1919). So the deviation of spacetime from homogeneity (due to the presence of a "heavy body" like the sun) is objectively observable and cannot be cured by pure conventions. Of course you can, according to the (weak) equivalence principle, always find a locally inertial (and thus isotropic and homogeneous) frame of reference but, at presence of non-negligible gravity, never a global one!

But, I didn't understand, why I should do that. Of course, you can in principle choose any kinds of coordinates, but how does it help to understand the determination of the speed of light a la Roemer?

It shows that using a model in which light speed is anisotropic, that the Roemer measurement would measure the two way light speed. That is, it would not exclude the ability to consider light speed highly anisotropic.

I'm glad we're getting back to the original question. One of the messages that the OP didn't have time to read points out that if you want to know two things - the one-way speed of light and the synchronization convention - you need to measure two things. Since there's only one measurement, I can pick a synchronization convention and get any one-way speed I like.