The context is live PA, not recorded.
So I deal with a singer that has a very full voice -- the power is all over the low mids. It's actually a great strong voice. There's a catch...
Her mic technique is a bit rubbish and so she needs a compressor.
Thence lies the problem. Her voice is already plenty thick enough without a compressor. using a compressor basically means that to get any diction or separation from the piano at all means taking so much out of her voice that it loses much of its characteristic.
Hence the question in the title. Yes, I want to have my cake and eat it.