Should the speed of light be slightly uncertain?

[fluidistic]

velocity as the derivative of position wrp to time is a low speed approximation.

I didn't know this. What is the general relativistic relation then? Or the not approximated expression?

If it were accurate at high speeds, we could accelerate things to light speed.

Why? I really don't see the implication.

In other words, V_total= V₁ plus V₂ is an approximation...

I know this but I don't see how it relates to "v=dx/dt implies v=v_1+v_2".

About the quotes you posted on the HUP, in fact I think the problem is that we aren't making any measurement. Consider any photon, you know it's moving at a speed "c" without any uncertainty. You don't even need to make a measurement for that, else special relativity is violated.

So let's say I measure the position of a photon with a screen that reacts to photon by darkening or something like that. I'd have a finite uncertainty in the position and also in the momentum (the HUP is not violated), but apparently none in the velocity. I'm still puzzled on how to get no uncertainty in the speed (that ok, I did not measure since I know it no matter what) but I have an uncertainty in the position.

Edit: I think I'm starting to understand something. The HUP is not related -directly at least- to the speed of a photon but on its momentum and position. Unlike a massive particle, the momentum of the photon is not related to its speed so an uncertainty in its momentum does not imply an uncertainty in its speed. Instead, an uncertainty in position should raise an uncertainty in velocity, not necessarily speed.
Am I wrong if I think that it's possible to have an uncertainty in position and in velocity and no uncertainty in the speed at the same time?
This would solve the problem...

Basically you know how fast the photon is moving no matter what. You don't know "perfectly" its momentum nor its position, all this at the moment it hit your screen. However there's an uncertainty in the direction the photon when it hit the screen.
So to answer the OP, the HUP principle applied to a photon does not imply an uncertainty in speed. Instead, in the velocity due to the uncertainty associated to its position.
Does this make sense? This does to me.