Are quasi-polynomial sized circuits for 3-SAT trivial?
If what you are studying worked out, it definitely would not be trivial.

It would imply that 3SAT has (non-uniform) circuits of size $n^{O(\log n)}$. Then, every language in $NP$ (and the polynomial time hierarchy) would have quasi-polynomial (i.e., $n^{O(\log^c n)}$) size circuits.

Even if it took $2^{2^n}$ preprocessing time to produce a data structure of size only $2^{O(\log^2 n)}$ which could then correctly answer arbitrary 3SAT queries of size $n$ in $2^{O(\log^2 n)}$ randomized time with high probability, 3SAT would have quasi-polynomial size circuits, using the known translation of randomized algorithms to circuits. This would not improve the known algorithm time bounds because of the preprocessing, but it would still be extremely interesting as a non-uniform result.

What do you mean by "to within an error that can be adjusted to an arbitrary amount"? Is the algorithm randomized?