I've got what I can only describe as a 'mooing' sound coming from the right side, under the SBC. I think it's the secondary air injection pump but I'm not 100% sure. Is there a way I can confirm that?

With a stethoscope you can hear with total clarity what each accessory is doing. Recommend you get one and start prodding around to isolate the fault.

The automotive type that is - the one with the long steel rod as the pickup.

Is it not possible to disconnect it by removing the electrical connector?

Don't know but I'll jack up the car tomorrow and take a peek. I'm guessing that if I unplug the connector I'll get some warning lights. But if I don't hear the sound, I'm in the right direction.

The air pump only runs at startup for a minute or so. Is that what you are hearing?
I believe there is a fuse for it. Pulling it is probably easier.
The only cow sound I can think of is the fuel pump and it may reverberate through the chassis.

Yep. About 30 seconds and then again about 5 minutes later. After that I won’t hear it again until it sits overnight.

I’ll take a look to see if I can find the fuse.

If you remove the fuse or disconnect the pump you will get a code.
If you are proficient with Star/DAS, you can disable the secondary air pump and thereby not get a code.

Here's what I did on DAS. I haven't driven it in a few days so I got a battery warning. I didn't want to do it without turning the car off so I did it with the car on. But I'll have to wait till tomorrow to see if this worked.
1. Select your car
2. Go to Control Units
3. Go to Drive
4. Go to MESFI 2.8
5. Go to Development Data
6. Go to Control Unit Adaptations
7. Go to Control Unit Adaptations(Variant Coding)
8. Go to Implizite Kodierung ME281 AMG C6-Stand
9. Scroll down to 0178 and change Diagnose ein Diagnose aus

I had a bad relay that controls the air pump. That produced a code. I was able to reset it with my Foxwell scanner.