At the suggestion of the original poster, I am summarizing an alternate answer that has a few strengths relative to my original answer. It is related to the question and answer at this MathOverflow Question.
For any matrix $A$, define $$ D_A(i,j)=\frac{A_{i,j}A_{i+1,j+1}}{A_{i+1,j}A_{i,j+1}} $$ and $$ L_A(i,j)=\log(D_A(i,j)) $$
Let $S$ be the set of $n\times m$ matrices with fixed row and columns sums $p_i$ and $q_j$ and positive entries. Let $Q:S\rightarrow R^{(n-1)\times (m-1)}$ be the map where the $(i,j)$-th entry of $Q(A)$ is $L_A(i,j)$.
Then $Q$ is a bijection between $S$ and $R^{(n-1)\times (m-1)}$. Moreover, the inverse is efficiently computable. Mpiktas wrote an R package called retacoro that appears to recover the inverse through gradient descent. Alternately, the inversion of this projection can be expressed as a geometric program, and geometric programs can be solved in polynomial time (and efficiently in practice).
Establishing the bijectivity of $Q$ and defining the geometric program precisely seem a bit long for a MathOverflow post. I'll post a more detailed description to ArXiv and put a link here when I'm done.