but the transformer produces noise through the speakers, even when the amp is turned off.
I suspect this isn't true. I do suspect that the transformer is noisy, even with no load. I would use a transformer with **at least** double the capacity needed with the amp at full output at the very least (this will prevent it from distorting the AC power sine wave). The transformer does not need to be rated for 'audio duty' (not sure there even is such a thing) but it does have to be quiet and there are ratings/specs for that although they are not always published.