A correspondent from Shakespeare's Globe writes to ask whether bottom ever meant 'posterior' in Elizabethan England. He has noted the way some modern productions make risque jokes about the character of Bottom in A Midsummer Night's Dream, and wonders if these are what Shakespeare intended. He wonders, too, whether Bum (the name of Pompey in Measure for Measure) would have had a similar connotation.
This is the kind of case where the amazing Historical Thesaurus of the Oxford English Dictionary comes into its own. Type bottom into the search, and up will come all the senses of that word, grouped thematically. Find the one which means 'buttocks', and there you will find a list of over 80 lexical items for this area of the anatomy, which you can see alphabetically or in chronological order of record. They contain items which are a mixture of learned, jocular, euphemistic, and slang.
1200s: cule, latter end, fundament, buttock
1300s: tut, tail, toute, nage, tail-end, brawn, bum
1400s: newscher, croupon, rumple, lend, butt, luddock, rearward, croup
1500s: backside, dock, rump, hurdies, bun, sitting-place, prat, nates, crupper, posteriorums,
1600s: cheek, catastrophe, podex, posterior, seat, poop, stern, breek, flitch, bumfiddle, quarter, foundation, toby
1700s: rear, moon, derriere, fud, rass, bottom
1800s: stern-post, hinderland, hinderling, ultimatum, behind, rear end, hinder, botty, stern-works, jacksy,
1900s: sit, truck-end, tochus, BTM, sit-upon, bot, sit-me-down, fanny, beam, ass, can, keister, batty, bim, quoit, rusty-dusty, twat, zatch, booty, bun, tush
So, bum? Yes, that would have carried a rude connotation in Shakespeare's day. But bottom? No. To exploit rude connotations here would be an anachronism. Of course, it's difficult to ignore the modern meaning, when we hear the name, but we have to try, if we want to get closer to Shakespeare's usage.