Need advice on a semi-classical derivation of the magnetic Schwinger limit

I'm trying to formulate with some rigor a semi-classical definition of the Schwinger limit on magnetic field strenght, but there are holes that need to be patched. Currently, the following formulation isn't satisfying and I need help on this.

Consider an electron dropped into an uniform static magnetic field. The initial velocity is perpendicular to the magnetic field lines. From Newton'equation (taking special relativity into account), the electron will follow a circular path of radius $$\tag{1} r = \frac{\gamma \, m_0 \, v}{e B} \equiv \frac{p}{e B}.$$ This relation shows that if the field $B$ is very strong, the radius will be small : $r \rightarrow 0$ when $B \rightarrow \infty$. For a given field, the radius will also be small if the momentum $p$ is small : $r \rightarrow 0$ when $p \rightarrow 0$.

The classical circular motion is well defined only when the uncertainties are relatively small : $r \gg \Delta r$ and $p \gg \Delta p$. The motion will be quantum mechanical (i.e circular motion not well defined) when the field is very strong and the momentum low enough : $r \approx \Delta r_{\text{min}}$ and $p \approx \Delta p_{\text{min}}$ (by symetry of the circular motion, I'm assuming $\Delta x = \Delta y \approx \Delta r$ and $\Delta p_x = \Delta p_y \approx \Delta p$, which I feel unsafe). So (1) could define the quantum mechanical regime : $$\tag{2} \Delta r_{\text{min}} \approx \frac{\Delta p_{\text{min}}}{e B}.$$ The Heisenberg uncertainty principle imposes a constraint on the value of $\Delta r_{\text{min}}$ and $\Delta p_{\text{min}}$ : $$\tag{3} \Delta r_{\text{min}} \, \Delta p_{\text{min}} \approx \frac{\hbar}{2}.$$ Substituting (2) into (3) give this contraint on the magnetic field, to get a quantum mechanical regime : $$\tag{4} B \approx \frac{2 \Delta p_{\text{min}}^2}{e \hbar}.$$ Currently, the uncertainty $\Delta p_{\text{min}}$ is arbitrary. The motion will be quantum mechanical as soon as $p \approx \Delta p_{\text{min}}$ and (4) are satisfied. The particle's motion will be quantum relativistic when $p \approx \Delta p_{\text{min}} \approx m_0 \, c$, so (4) becomes $$\tag{5} B \approx \frac{2 m_0^2 \, c^2}{e \hbar},$$ which is the Schwinger limit (up to a factor of 2). For magnetic fields stronger than this, the particle's motion will be ultra-relativistic.

Now, there's something fishy with this derivation that I don't see clearly. What is wrong with this reasoning ? Where are the holes in the derivation shown above ? Also, is there a simpler way of getting (5), with some rigor in the reasoning ?

The problem may lies in the physical interpretation and wordings, not just in the maths (which are easy).

EDIT : It's simpler and more accurate (?) to use the angular momentum instead of Heisenberg principle : $$\tag{6} L = r \, p = \frac{p^2}{e B} \approx \hbar.$$

The uncertainty principle you used is for one dimension, but I believe the orbit of the electron must be defined in two. If you are assuming $$\Delta x$$ is equal to $$\Delta y$$, then they both must equal $$\Delta r/\sqrt{2}$$. The same would go for $$p$$, so in the situation, you are describing: $$$$\frac{\Delta r_{min}}{\sqrt{2}}\frac{\Delta p_{min}}{\sqrt{2}} \approx \frac{\hbar}{2},$$$$ so then $$$$\Delta r_{min}\Delta p_{min}\approx \hbar.$$$$ Following this through to the end of your derivation yields the correct limit.