Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations in variables $m',m,n,k_1,k_2,k_3$
$$m'm^2=1+k_1p^{t-2}q^{t}$$ 
$$m'n^2=1+k_2p^tq^{t-2}$$
$$m'mn=1+k_3p^{t-1}q^{t-1}$$
where $m,n$ are coprime with $(mn,pq)=1$ and $m,n=\Theta(\max({p^{(t-1)}},{q^{(t-1)}}))$.

Essentially given coprime $p,q$ and integer $t\geq3$ is there coprime $m,n$ with $(mn,pq)=1$ and $m,n=\Theta(\max({p^{(t-1)}},{q^{(t-1)}}))$ such that there is a common integer $m'$ serving as modular inverse of $m^2\bmod p^{t-2}q^t$, $n^2\bmod q^{t-2}p^t$ and $mn\bmod p^{t-1}q^{t-1}$?

From counting arguments it looks like there is room for $m,n=\Theta(\max({p^{2(t-1)}},{q^{2(t-1)}}))$? Can we replace $2$ by anything in $[1,2)$?