Suppose $n$ is an integer with factorization $n = ab$, and $a, b$ unknown and not necessarily prime.
From this point forward, assume that we are free to choose $p$.
Let $g$ denote a primitive element of the finite field $\mathbb{F}_p$. Therefore, $a = g^A, b = g^B$ for some $A,B \le p-1$. Without loss of generality, let $A \le B$ (in $\mathbb{Z}$). We can drop the equality if $n$ is not a perfect square in $\mathbb{Z}$.
Q1: Is it possible to choose $p, g$ such that $A$ is small relative to $p$?
Q2: Since there are $\phi(p-1)$ primitive elements in $\mathbb{F}_p$, an argument could be made that some primitive element $\gamma$ may yield $a = \gamma^\alpha$ where $\alpha$ is small relative to $p$, so the answer to Q1 is "probably yes". Suppose we choose a bound $\beta$, could we make a counting argument to estimate the probability that a randomly chosen primitive element $\gamma$ yields $a = \gamma^\alpha$ where $\alpha$ is small relative to $\beta$?
Q3: There are primes of special form where computing discrete logs is easy (i.e. polynomial time). For eg: Fermat Primes and primes where factors of $p-1$ are reasonably small where we could use Silver-Pohlig-Hellman discrete log algorithm. While $F_4 = 65537$ is the largest known Fermat prime and may not be suitable for large $n$ for our purposes, we are free to choose primes where factors of $p-1$ are reasonably small. For eg: primes of the form $2^mk +1$. It is known that there are infinitely many primes of this form.
Therefore, given a prime of the form $p = 2^mk + 1 > n$ can we find a primitive element $g \in \mathbb{F}_p$ such that it splits the two cofactors of $n$ into disjoint ranges containing consecutive powers of $g$ with $a \in [g^0,g^\beta]$ and $b \in [g^\beta, g^{p-1}]$ with $a = g^A$ and effectively searchable by brute force of the range $[g^0,g^\beta]$?