Thought experiment in relativistic quantum mechanics?

Background

Consider the following thought experiment in the setting of relativistic quantum mechanics (not QFT). I have a particle in superposition of the position basis:

$$ H | psi rangle = E | psi rangle$$

Now I suddenly turn on an interaction potential $H_{int}$ localized at $r_o = (x_o,y_o,z_o)$ at time $t_o$:

$$
H_{int}(r) =
begin{cases}
k & r leq r_r’
0 & r > r’
end{cases}
$$

where $r$ is the radial coordinate and $r’$ is the radius of the interaction of the potential with origin $(x_o,y_o,z_o)$

By the logic of the sudden approximation out state has not had enough time to react. Thus the increase in average energy is:

$$ langle Delta E rangle = 4 pi k int_0^{r’} |psi(r,theta,phi)|^2 d r $$

(assuming radial symmetry).

Now, lets say while the potential is turned on at $t_0$ I also perform a measurement of energy at time $t_1$ outside a region of space with a measuring apparatus at some other region $ (x_1,y_1,z_1)$. Using some geometry it can be shown I choose $t_1 > t_0 + r’/c$ such that:

$$ c^2(t_1 – t_0 – r’/c)^2 -(x_1 – x_0)^2 – (y_1 – y_0)^2 – (z_1 – z_0)^2 < 0 $$

Hence, they are space-like separated. This means I could have one observer who first sees me turn on the potential $H_{int}$ and measure a bump in energy $langle Delta E rangle $ but I could also have an observer who sees me first measure energy and then turn on the interaction potential.

Obviously the second observer will observe something different.

Question

How does relativistic quantum mechanics deal with this paradox?