Is Type I lattice trapdoor hard to find even given oracle access to compute inverse of trapdoor function?
If the adversary is a classical algorithm, then the answer  to your question is not known. But if the adversary is a quantum algorithm that can query the oracle in superposition, then the answer is yes: by making queries to the oracle on certain (efficiently produceable) quantum states, it can recover a Type-I trapdoor for $A$.

For classical algorithms, the difficulty is that we do not know how to generate a valid output of $g$ (that the oracle is obliged to succeed on) except by choosing $s,e$ ourselves and feeding them through $g$. But then the oracle’s answer is useless to us, because we already know the answer before querying it.

For quantum algorithms the situation is different. We can prepare a certain superposition over “all” valid $s,e$, feed it through $g$ to get a superposition over outputs, and invoke the oracle to get back $s,e$. Critically, this allows us to uncompute (or “forget”) our original choices, leaving us with a useful quantum state. Specifically, by taking its quantum Fourier transform we get a superposition over short vectors $x$ such that $Ax=0$. By measuring the state we get such an $x$, and can repeat to accumulate many linearly independent ones, yielding a Type-1 trapdoor.

The difficulty in the classical setting, and the quantum strategy for dealing with it, are from Regev’s original paper on LWE. The connection to SIS/LWE was first described in the GPV paper.