Product measure and its marginals
No. Let $X=[0,1]$, $m$ be the uniform distribution on the diagonal $D=\{(x,x)\mid x\in [0,1]\}$, and let $\lambda$ have the constant value $1$. The question then reduces to whether the uniform distribution on the diagonal is proportional to a product measure. It is not. Indeed, the marginal distributions of $m$ are the uniform distribution on $[0,1]$, but the corresponding product measure assigns measure zero to the diagonal.