So it is a battery analyser. That means the output swing (in voltage) is actually very low and so is the power. I think you are going at this the wrong way entirely. I have designed/made similar devices (some that needed to go up to 300VDC) and I always used a capacitor in series with the current source to get rid of the DC offset (with a fast settling circuit at startup). That way you can use a simple but precisely controlled linear current source.
I tend to agree, this is best designed as a current amplifier, not a voltage amplifier and could even be done as a 4 terminal system that uses a series resistor 'dummy load' approach to avoid the need to try to drive
millivollts into milliohms just to try to get your Stiff-DC-Source, into the teensy transfer zone. ie Even if you can do that, the noise-floor will dominate the current result.,
A 4 terminal system means you need 2 good ADCs and a current meter, tho that could be a 3rd ADC across the known and stable series resistor.
A quick Digikey search finds
CSR2512C0R05F Riedon RES 0.05 OHM 1% 3W 2512 16,000 - Immediate $0.18354/2,000 50 mOhms ±1% 3W ±50ppm/°C
5A into 50mOhms is 1.25 Watts and needs 250mV of swing (and that lands you in some nice current-ADC specs, high volume parts)
My concern of the capacitor wasn't so much ESR, but it's actual value. For it to have let's say 10V of voltage drop at 1mHz and 5A, the required capacitance would be ~ 80 Farads.

yes, OUTPUT end AC coupling gives big numbers, but it does not need to be applied at the output pin.
I'm not sure a cap is needed, as you can build a current amplifier and manage the voltage swings that way. The test software can emulate AC coupling as needed.
Having a finite, series R helps greatly in all of this.