PeterDonis said:
Which means that the "proper velocity", as you are calling it, is not an invariant (because it's calculated from a frame-dependent quantity), so it's (a) misnamed (the adjective "proper" is usually reserved for invariants) and (b) not physically meaningful.
I don't see how that can be true if it's a frame-dependent quantity, per the above.
There's a 1:1 mapping from this quantity to the 4-velocity, which is of course manifestly frame independent. By the way, I got the term "proper velocity" from a paper which discussed various notions of velocity in SR, namely proper velocity, celerity, and rapidity. I forget the name, but I'm sure I could dig it up if needed. There's some small but probably non-zero possibility that my memory of the terminology could be slightly off.
To give a specific but coordinate dependent description of this procedure, if we use coordinates t,x,y,z, then the quantity in question can be written as dx/dtau, dy/dtau, dz/dtau. To get the fourth component and find the 4-velocity, we do need to compute the fourth component of the 4-velocity, dt/dtau from the other components using the rule that -dt/dtau^2 + dx/dtau^2 + dy/dtau^2 + dz/dtau^2 = -1, i.e. the fact that the length of the 4-velocity is -1 (using my usual choice of sign conventions and using units where c=1). But this is easily done.
Note that the 4-velocity also gives us the 4-momentum for a point particle.
So, I think I've demonstrated that there is a meaningful notion of velocity without worrying about clock synchronization. Clock synchronization has other uses, but we don't need two clocks to have some concept of velocity, we can easily make do with one and avoid the whole mess in the case of physical objects which travel timelike worldlines..
I haven't seen this discussed anywhere in detail.
The thought experiment that set me down this path ages ago was considering airplanes and time-zones. I'm idealizing the experiment to avoid complicating factors such as prevailing winds and wind resistance.
The trip time, using wall clock (which are set via local time zones), depends on the direction of travel. The same airplane travelling west-east and east-west take a different amount of time to transverse the same path. Do we take seriously the idea that the airplanes have different velocities depending on their direction of travel? Would we claim that they had different momenta? In general, we do not, it's pretty obvious it's just an artifact of how we chose to synchronize the clocks. And there's an very easy and straightforwards way to tell that the travel time of the (idealized, no-wind) airplane really doesn't depend on the direction. Just put a clock on the plane and notice that the elapsed time doesn't depend on direction.