I naively thought that given my vocal range I clearly ought to pick out the bass part from the sheet music and sing that.

So you should; but it takes quite a lot practice to sing part music at sight. If you've never done it before, you shouldn't be too surprised if it turns out to be tricky. There's no better way to get practice than to join a choir, of which there are many around here.

Unfortunately, singing bass appears to be incredibly hard

In four-part harmony, bass is usually the second easiest part to sing (after soprano): you have the fundamental note of each chord. Tenor and alto are often much harder, because they often don't make much sense as tunes (though a skillful harmonist, like J. S. Bach, can make all four parts interesting).

the men around me seemed to be singing the soprano part transposed down by however many octaves seemed appropriate

That's the usual thing to do if you can't sing in parts.