Generating Graphs with Trivial Automorphisms
At least the first naive approach one might think of does not work.  The approach I have in mind is to simply generate $G_1$ purely at random.  Since almost all graphs have no symmetries (that is, the proportion of graphs on $n$ vertices with no nontrivial automorphisms approaches 1 as $n \to \infty$), $G_1$ will have no nontrivial automorphisms with high probability, which is what we want.  However, the average-case version of graph isomorphism, where graphs are chosen uniformly at random, can be solved in linear time [BK], so this does not generate a distribution of hard instances.

But a second naive approach has a chance of working: generate a random regular graph (of non-constant degree, since constant-degree graph isomorphism is in P).  This also has no nontrivial automorphisms with high probability [KSV], but the Babai-Kucera result does not apply (as they point out in the paper).  Proving that this is an invulnerable generator obviously requires some assumptions, but one could imagine proving unconditionally that average-case regular graph isomorphism is as hard as worst-case graph isomorphism, though I don't know how likely this is.  (Note that worst-case regular graph isomorphism is equivalent to worst-case (general) graph isomorphism.)

[BK]. Laszlo Babai, Ludik Kucera, Canonical labelling of graphs in linear average time. FOCS 1979, pp.39-46.

[KSV] Jeong Han Kim, Benny Sudakov, and Van H. Vu. On the asymmetry of random regular graphs and random graphs. Random Structures &amp; Algorithms, 21(3-4):216–224, 2002.  Also available here.