Experiments included Lil Gun in the 357 and lubes in the '06.
20 gr. Lil Gun, CCI #500 primers   1159 fps, 135 ES, 8.8" horizontal dispersion.
20 gr. Lil Gun, Fed #200 primers   1160 fps, 159 ES, 8" horizontal dispersion.
Conclusions   This load was highly compressed yet still gave low velocites.   To heck with Lil Gun.
I tried something different today, using three 3-shot groups, plus the aggregate of the 9 shots, instead of three 5-shot groups.   The idea is to get more information with fewer shots.
58 gr. IMR4350, 0.312", Speed Green lube   average 2.55", 2742 fps. 21 ES, all 9 shots 3.6" 33 ES.   Even though it started with a clean, cold barrel, the velocity and group size remained consistent for all 9 shots.   I was not expecting too much from this soft lube but it surprised me.
58 gr. IMR4350, 0.312", Rooster HVR   average 2.70", 2749 fps. 27 ES, all 9 shots 4.3" 57 ES.   Rooster gave the highest velocities, just like it did in the 357.   It appears to be the cleanest lube with this load, though that is subjective.   7 shots were in 2" so there is some potential here.
61.5 gr. RL 19, 0.312", Ballisticast lube   average 1.97", 2727 fps, 32 ES, all 9 shots 3.15", 73 ES.
58 gr. IMR4350, 0.312" x 0.302", Ballisticast lube   average 6.5", 2713 fps, 44 ES.   This bullet had been nose-sized in a tapered die so that it could be seated out farther.   The lower velocity is due to the different seating depth.   I stopped after 6 shots because it obviously was not going to shoot well.
58 gr. IMR4350, 0.309", Ballisticast lube   average 2.85", 2708 fps, 18 ES, all 9 shots 4.7", 38 ES.   The 0.309" bullet gave up 50 fps, perhaps due to poor gas sealing.   The velocity seemed to drift downward as more shots were fired, though in fairness, that has been normal with Ballisticast lube which just doesn't seem adequate in this load.
Conclusions   Thumbs up on Speed Green, Rooster, and RL 19.   Thumbs down on the tapered bullet, and thumbs down on Ballisticast lube (I was hoping I could use Ballisticast because my 357 likes it).  Undecided on optimal bullet diameter.   I am pretty happy with this bullet, and there still are many variables to tweak that might make things better yet.