Can we verify satisfiability of first order statements via saturation in sub-exponential time?
I am assuming by saturation you mean saturation-based reasoning as used by a resolution theorem prover e.g. where we negate and attempt to find a contradiction by saturation if we saturate we can conclude there is no contradiction assuming certain properties were preserved during proof search.

Given this setting, it is the part in bold that kills any attempt to do what you are hoping for. To be able to conclude that a saturated set was produced using a complete proof search procedure means checking every inference not made during proof search to make sure this was allowed. In practice this means looking at selected literals of every given clause and all reductions or deletions.

As far as I'm aware, so far there has been no effort to make theorem provers record the necessary information to check these things. But such information could be exponentially larger than the saturated set.