Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and an integer $m$ is there a polynomial in $n$ algorithm to find a $\{-1,0,+1\}$ matrix $T$ such that $$\mathsf{det}(L+T\odot M)=m$$ where $\odot$ refers to $ij$th entry of $T\odot M$ being $T_{ij}M_{ij}$?

Is there any canonical approach to this problem that can beat $2^{O(n)}$ time?

An evidence $2^{O(n)}$ could not be beaten would come if we show that 'Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and integers $m,t$ is there a $\{-1,0,+1\}$ matrix $T$ with $\|T\|_F\leq t$ such that $\mathsf{det}(L+T\odot M)=m$ holds?' is $NP$-complete (it is in $NP$

~~however there seems no canonical reduction from any $NP$ complete problem and I find every theoretical support for an $NP$ complete reduction seems to turn to absurdity here~~after the presented answer I think it should be $NP$ complete)?

~~It seems in all likelihood this problem is in $P$~~.

If $T$ is a planar bipartite signed biadjacency then the case could be special and might be handlable within $P$. This possibly yields difficulties in reductions.