Which commercial shotshells are preferred for sporting clays in 1930s vintage 12 ga. LC Smith doubles?

If the head of the stock has been acraglas bedded 1 1/8 oz at 1200 fps target loads are reasonable.
If not, in light of the age of the wood, I'd suggest 1 oz. at 1200 fps.
I use B&P Comp One 7/8 oz at 1160 fps for skeet. Fiocchi also makes a 7/8 oz. target load.

I agree with Dr. Drew. I don't think I've ever shot 1 1/8 loads in my 12 ga. Smiths. Keep it at or under 1,200 ft./sec. with 7/8 oz. or 1 oz.
Check screws regularly, especially the tang screws (one under the top lever, and one under the trigger guard).
Note that 7/8 oz. of 8's actually has a few more pellets than 1 Oz. of 7 1/2's. Unless you expect targets over 35 yds., 1 oz. isn't needed.