If you move your rear wheel as far back as possible in the drops, does it increase clearance a bit?

Frankly, some tires are simply measured incorrectly. Racing oriented tires tend to be undersized so they can brag of being lighter than they really are, while a tire like an Armadillo will run quite large. Also, some tires seem to have a rounder profile or more V-shaped profile, compared to flatter tires (which are sometimes a bit wider). In any case, I wouldn't expect any 35c tire to fit an old road conversion.

Do not remove the rear brake bridge- but you might try filing down the "washer." It doesn't provide any structural support and it will give you a few more millimeters.