Saving file as JPEG with CMYK colors come out muted to the Printer

I pulled in a design, that was Pantone. I changed all to RGB, only because I thought printing sublimating advertising flags is usually in RGB. The company printing the advertising flags, want it in CMYK as a JPEG. So I used the "Find & Replace" feature and change all colors to CMYK. When I save it as JPEG and change to CMWK in the "Export to JPEG" box, even though the colors look bright, when I send to the Vendor that is going to print the flag banners, the colors are all muted. Oddly, when I pull into Corel Draw as a JPEG, it is bright. 

What am I supposed to be checking in the JPEG export box, so colors stay bright. I have tried everything, and I do not get it. Especially since it imported into my Corel Draw brightly. There is obviously a setting that I should be clicking to make it stay bright for them.

I so appreciate the help!