Inconsistency arising from a potential function that depends on time? In a 1d setting, suppose we have the following potential:
$V(x,t) = f(t)$
for some function $f(t)$. Notice that this potential does not depend on position $x$, it only depends on time $t$.
It follows that a particle would not feel any force: $\nabla_x V(x,t) = 0$.
Now we do a Lorentz boost with velocity $v$. The relation between the original time coordinate $t$ and the new space-time coordinates $x',t'$ is given by the inverse Lorentz transformation $t=g(x',t',v)$. It follows that the potential, in the new coordinates, is $V(x',t')=f(g(x',t',v))$. Now we see that the potential does depend on position, and the particle does feel a force!
That the particle feels a force in one inertial frame but doesn't feel the force in another inertial frame is an error, I believe. (for example, it isn't consistent that the velocity of the particle is constant in one frame but not in the other). How is this inconsistency resolved?
Now, obviously, what matters is not potential values, but potential value differences (at least, at equal times). So we could have ignored the $f(t)$ term altogether. But, are we not allowed to include this term?
Also, is there a general recipe for "ignoring" the time dependence? when the time dependence is a simple additive term (as in our example), the recipe is clear. But what if the dependence on time is not separable from the dependence on position?
 A: 
That the particle feels a force in one inertial frame but doesn't feel the force in another inertial frame is an error, I believe. (for example, it isn't consistent that the velocity of the particle is constant in one frame but not in the other). How is this inconsistency resolved?

It is neither an error nor an inconsistency.
Recall that in relativity energy and momentum are joined into the four-momentum $P^\mu=(E,\vec p)$. Similarly, force and power are joined into the four-force $f^\mu=(P,\vec F)$. The four-force is the rate of change (with respect to proper time) of the four-momentum $f^\mu = \frac{d}{d\tau} P^\mu$.
So a four-force that has only a time component, $P$, in one frame is expected to have a space component, $\vec F$, in other frames. That is how relativity works. What is only energy in one frame is also momentum in another. Therefore what is only a change in energy in one frame is also a change in momentum in another. Therefore what is only a power in one frame is also a force in another.
In this case, in the frame where there is only power the energy is increasing and by $E=mc^2$ the mass is also increasing, the velocity does not change. In the other frames the force also does not change the velocity. Instead, it provides the increased momentum needed for the more massive object to continue moving at the same speed.
There is no inconsistency, simply an object that is changing mass due to the energy that is being added.
