PeterDonis said:
No. That claim is the claim that ##\vec{a}' = \vec{a}##.
The claim I'm questioning is the claim that the relationship between ##\vec{a}## and the fields ##\vec{E}## and ##\vec{B}## and the velocity ##\vec{v}##, which is what the Lorentz Force equation expresses, is the same in both frames. That claim is a claim about the relationship between accelerometer measurements, velocity measurements, and field strength measurements. It's not a claim about accelerometer measurements alone.
As I said before, it's much simpler to study the exact special-relativistic equation. There's no merit in the non-relativistic limit here at all. Of course you can get easily the non-relativistic limit in the usual way:
The covariant equation reads
$$m \ddot{x}^{\mu}=\frac{q}{c} F^{\mu \nu} \dot{x}_{\nu},$$
where dots denote derivatives wrt. proper time, ##m## the invariant mass of the particle (scalar), ##q## the charge of the particle (scalar), and ##F^{\mu \nu}## the components of the Faraday tensor of the given field.
First of all, not all four EoMs are independent since, as it must be the equation is compatible with the constraint, following from the definition of proper time
$$\dot{x}_{\mu} \dot{x}^{\nu}=c^2 =\text{const}.$$
Thus it is sufficient to consider the spatial components only, and that reads very familiar when expressed in the (1+3)-form of the equation in the given reference frame:
$$m \ddot{\vec{x}}=q \left (\vec{E}+\frac{\dot{\vec{x}}}{c} \times \vec{B} \right).$$
Although this looks like the non-relativistic equation, it's not, because the dot denotes a derivative wrt. to proper time, ##\tau##, rather than "coordinate time" ##t##. Nevertheless this equation is form-invariant unter Lorentz transformations since it's derived from a equation only involving tensor components.
Then, in any frame, where the non-relativistic approximation is valid, i.e., in a frame, where (during the considered time!) at all times
$$\left |\frac{\mathrm{d} \vec{x}}{\mathrm{d} t} \right| \ll c.$$
Then you can approximate
$$\dot{f}=\frac{\mathrm{d} f}{\mathrm{d} t} \frac{\mathrm{d} t}{\mathrm{d} \tau} = \mathrm{d}_t f \frac{1}{\sqrt{1-(\mathrm{d}_t \vec{x})^2/c^2}} = \mathrm{d}_t f [1+\mathcal{O}(v^2/c^2)]$$
with ##v=|\mathrm{d}_t \vec{x}|##.
Thus, using the correct relativistic equation and the correct transformation between frames (Lorentz transformation) leads to the conclusion that also the non-relativistic approximation is forminvariant in all frame it applies. I don't know whether the somewhat cumbersome calculation with Galilei transformations here really helps. In any case you have to take the corresponding limit of the transformation law for the field components too. The above covariant approach, however, suggests that this should lead to the same conclusion if done properly (i.e., taking consistent approximations in powers of ##v/c## on both sides of the equation).