Kama mashma'uyot yesh la'milah ha'zot?
"How many meanings are there for this word?" should be acceptable too, right? Or would that need בשביל ?
Yes of course your translation should be acceptable, it's perfectly normal English. בשביל suggests you are doing something "for" somebody's sake or it's a present "for" them; you wouldn't use it here.
זאת שאלה טובה לדעת בעברית!
How about "This word has how many meanings?"
The word בר has at 6–7, depending on how you count.
(I'm amazed at the downvotes. What did they expect, but hey) Nice! By my count: ערב nine; חבל חלל seven; אבל בר חרש six; אל מטה משל five; מן רע four; culling the most diverse.