If $r\in\Bbb Z_{\geq0}$ and $m$ is odd then let $2^\ell\mid\binom{m}{2^r}$ and $2^{\ell+1}\nmid\binom{m}{2^r}$.

Is there a way to find if $\ell$ is even or odd without computing $\binom{m}{2^r}$ (assume $\ell$ odd if $m<2^r$)?

Lucas theorem states $\ell$ is number of carries when $m-2^r$ is added to $2^r$.

However we look for less information and may be we can get away from explicit adding and using other type of operations.