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10 ohms in series with a capacitor is not a short circuit at high frequencies. Instead it is 10 ohms at high frequencies.
A transformer works very poorly (low amplitude and lots of phase shift) at the very low frequency that you say is "slow".
A class-D audio amplifier IC is smart. It has current limiting and high temperature shut-down.
But for your electrical application you probably want to select the amount of voltage produced by your current transformer and its shunt resistance.
So select a class-D amplifier IC that produces more current than you need and connect your shunt resistor to its negative feedback input if it has one.