Your expressions are off, as is your characterization of BJ! I will use the correct ones, found in most texts, e.g. this. All orderings, including these ones, differ only by terms of order $\hbar$, resulting from application of the commutation relations.
$$Q_{Weyl} :p^sq^r\rightarrow \frac{1}{2^s}\sum_{k=0}^{s} \binom{s}{k}\hat
p^{s-k}\hat q^r\hat p^k\\
Q_{B-J} :p^sq^r\rightarrow \frac{1}{s+1}\sum_{k=0}^{s}\hat p^{s-k}\hat q^r\hat p^k.$$
Weyl's prescription is the most symmetric one, as it weighs every possible ordering with multiplicity one, given its generating function. Namely, the coefficient of $\tau^s ~\sigma ^r$ of $(\tau \hat p + \sigma \hat q )^{s+r}$ is the above Weyl ordering, whose ease of manipulation is self-explanatory in phase-space quantization, as explained in books. The compact expression you have, above, is a condensation due to McCoy (1932), preferred only by mathematical dandies.
Check that, in Weyl ordering, as stated,
$$
Q_{Weyl} : \bbox[yellow]{ 6p^2q^2\rightarrow \hat p ^2 \hat q^2 + \hat q^2 \hat p ^2 + \hat p \hat q \hat p \hat q + \hat q \hat p \hat q \hat p + \hat p \hat q^2 \hat p + \hat q \hat p ^2 \hat q\\ =\frac{6}{4}( \hat p ^2 \hat q^2 +2 \hat p \hat q^2 \hat p + \hat q^2 \hat p ^2 )}~.
$$
Further check that, if you are determined to use the compact expression, then the BJ prescription is superficially simpler in the McCoy systematics employed,
$$
Q_{B-J} : 3p^2q^2\rightarrow \hat p ^2 \hat q^2 + \hat p \hat q^2 \hat p + \hat q^2 \hat p ^2 ,
$$
until you look at every possible ordering sequence, or you try to do a real deformation quantization calculation where real answers are sought.
