There are many analogies between $\mathbb F_q[x]$ and $\mathbb Z$.

Supposing there is a fixed (say $10^9$) dimension linear integer program (without any objective function) which assists in a deterministic polynomial time algorithm for factoring integers would it say anything about derandomization of $\mathbb F_q[x]$ factoring (the most important step is the case of equal degree factoring https://www.csa.iisc.ac.in/~chandan/courses/CNT/notes/lec8.pdf)?

Fixed dimension integer programming is known to be in $P$ by a result of Lenstra (https://www.jstor.org/stable/3689168).

To be clear assume that the integer program has no objective function and produces the factors in one of its variables. We have to set the program, run Lentsra's algorithm and determine the value of a particular variable for a non-trivial possibly composite factor by computing $GCD$ of this variable with the given integer externally to the program after the program is completed. We can reuse the program to completely factor the given integer with the help of $AKS$ algorithm (https://en.wikipedia.org/wiki/AKS_primality_test) for primality.

Essentially would such a set up help over $\mathbb Fq[x]$?