Andy,
It's because under Canon's system, Av resorts to 'fill flash' mode. Often people just assume that flash means 'fast shutter speed', but that's not always the case. With Av mode, the flash by default will try to just 'fill in' some extra light to make sure your foreground subjects are properly exposed. It will still however expose to ensure the background is also decently exposed. For this reason, you'll notice that the shutter speed doesn't actually differ compared to Av mode without using flash.
So just to recap, because Av mode's default flash paradigm is 'fill-flash', that means it cannot just raise the shutter speed like you want. Because if it did, the background may become poorly exposed, hence defeating the purpose of the mode being fill flash only. If you didn't care for fill flash, and just wanted the foreground subjects to be decently exposed (with fast shutter speeds in poor lighting), then use Tv like you have been or use Manual. Or, like already mentioned, there's a C.Fn available to force the shutter speed to over 1/200th second under Av mode. In other words, effectively disable it's fill-flash predisposition.