No Personel experience but this is interesting problem. A few options that spring to mind though.

Intra-cellular FACS staining after a set treatment period after treatment with il-10 and then a secretion inhibitor: You might see more IL-10 positive cells but I'm unsure if cells not treated il-10 would stain differently.

Alternatively you could us a secretion inhibitor followed by some other sort of detection method(ie add secretion inhibitor, wash cells, lyse and look for more IL-10 in the treated group via your method of choice)

You could treat your cells for a set period of time and then change the media to media without recombinant Il-10 then perform an standard ELISA.

You could blank your ELISA with empty well with only recombinant Il-10 but this introduces a lot of noise you would have to assume the secreted Il-10 is significant enough to overwhelm the noise from the blank wells.

You could look for expression changes in Il-10 mRNA via qPCR though this only shows the pathway is activated not that protein is secreted.

Measurement of an alternative marker of Il-10 production you may be able to use a protein induced by Il-10 to make the above measurements more convincing.

Militenyi sells very expensive kit. The workflow basically has you replate the cells so I personally would try replating and then running an ELISA first. The major advantage is that it in theory specifically stains only the secreting cells you just have to makes sure the cell binding side of the antibody dimer is expressed on your cells(I assume they use CD45 but I'm unsure). I imagine the optimization for this would be intensive. Finally any residual Il-10 would probably cause a false positive so you would have to monitor this somehow.

https://www.miltenyibiotec.com/US-en/products/macs-cell-separation/cell-separation-reagents/microbeads-and-isolation-kits/cytokine-producing-cells/il-10-secretion-assay-cell-enrichment-and-detection-kit-pe-human.html#130-090-435

Intracellular staining may or may not work since you are adding recombinant IL-10. Since the intracellular staining doesn't distinguish between produced and uptaken.

It MAY BE possible that you might find an antibody pair that recognizes the recombinant and not the endogenous one... But this requires you to screen and design your own assay.

I'd say just do gene expression of IL-10.

I agree with the gene expression recommendation. Alternatively, you could add IL-10 to one of the wells without the cells and subtract the values you got in your control from the experimental ones.