The Dell UC350 headset plugs into a USB controller which has a mic mute switch. Is this the part you called a "splitter"?
If so why do you need to add a switch?

Hi, the customer does not want to use the USB controller on this version. The splitter is this https://www.startech.com/en-gb/cables/muyhsfmm. This cable plugs into this rather than the USB controller.

 

Thanks for you help. I tried the circuit using this cap GRM31CR61E226KE15L and I am unable to get any sound out on the PC now. Any ideas?

In that case it is possible that it is an electret microphone and then @Audioguru again circuit with a bigger capacitor may do the job.

 

In that case it is possible that it is an electret microphone and then @Audioguru again circuit with a bigger capacitor may do the job.

Yes, it seems to be working with a larger capacitor. Now I just need to convince the customer to change the switch to a NC one!
Thanks for your help.

 

Hi, I tested this circuit and it does work but the mute can be overcome by talking louder through the microphone. Do you think that could be stopped by altering the values?

The circuit I posted is for an electret mic but maybe your mic is a completely different dynamic type.
Also, computer mic input has DC power for an electret mic but a dynamic mic must NEVER have the DC power.