Given $m\in\Bbb N$ large enough is there a coprime pair $a,b$ in $[2^{m-1},2^m]$ and $m\_1,n\_1, m\_2,n\_2\in[0,m^{c'}]\cap\Bbb Z$ with each of 
$$m\_1a-n\_1b,m\_2a-n\_2b\in[0,m^c]\cap\Bbb Z$$
$$m\_1n\_2\neq n\_1m\_2$$
true at some $c,c'>0$ independent of $m$? 



Existence of certain coprime numbers
What you seem to request is that the matrix $M=\begin{pmatrix}m\_1 & -n\_1 \\
m\_2 & -n\_2\end{pmatrix}$ were non-degenerate, and the vector $d:=M\binom ab$
had its coordinates of size polynomial in $m$. If this could be arranged, as
a result we would get $\binom ab=\Delta^{-1}Ad$, where $\Delta:=\det M$ is a
non-zero integer, and $A$ is a matrix with integer entries; moreover, $\Delta$
and the entries of $A$ are of size polynomial in $m$. But this would imply
$\Delta\binom ab=Ad$, which is impossible as the entries of the vector in the
left-hand side are of size exponential in $m$, while the vector in the
right-hand side has entries of size polynomial in $m$.


