Pick a large enough prime $P$ and a small $\epsilon>0$. Fix coprime $A$ and $B$ of size $P^{\frac13+\epsilon}$ roughly where $AB=P'-P$ holds at some prime $P'$.
Set $\alpha P'\equiv A^2\bmod P$ and $\beta P'\equiv B^2\bmod P$. Note this implies $\alpha B^2-\beta A^2\equiv0\bmod P$
Find $M,N$ of size $P^{\frac23+2\epsilon}$ at most such that $P|(\alpha B N - \beta A M)$ and $(MN,AB)=1$.
Is there $m,n$ such that if $$r\equiv\frac{(Mm+Nn)}{(A m + B n)}\bmod P$$ then either $r$ or $P-r$ is of size at most $P^{\frac13+2\epsilon}$ and $(\alpha B m + \beta A n)^{-1}\bmod P$ exists?
Note $r\equiv\frac{(Mm+Nn)}{(A m + B n)}\bmod P$ is same as $(M-rA)m\equiv n(rB-N)\bmod P$.
So naively taking $n\equiv (M-rA)\bmod P$ and $m\equiv(rB-N)\bmod P$ would give $(\alpha B m + \beta A n)\equiv (\alpha B (rB-N) + \beta A (M-rA))\equiv r(\alpha B^2-\beta A^2)\equiv0\bmod P$ and so $(\alpha B m + \beta A n)^{-1}\bmod P$ does not exist.