The capacitor is going to be in parallel with the output, not in series with it. It works together with the inductor to form a low pass filter rejecting the high frequency switching of the class D amplifier and only letting the audio pass through.
The amplifier output is in effect already DC-coupled, and the differential output makes it possible to avoid DC at the speaker while using a single power supply rail.
The TPA3116 also has integral DC protection, which will shut down the amplifier if DC is detected at the output.
Unfortunately, this is just the wrong IC for the job. About the only way you're going to do what you want is to float the entire power supply above ground potential (for example, the GND of the amplifier is actually +12V, and the VCC is at +24V, so you have 12V supplying the amplifier but the output always has 12V of DC on it). Such an approach would require of course a more complex power supply setup and will also require the input signal's ground to be DC isolated (by use of a transformer or capacitors), not a very practical approach.