asummers

Thanks, Walt. Reading up on and learning more about retaining/preserving color values. This is my first foray into printing with increased accuracy. 
I appreciate your input! -A

Thanks for your responses Walt and Hangman. 
Hangman: Thanks for identifying the issue  with U.S. Web Coated (SWOP) v2 and Generic CMYK. I have addressed that. I have been exporting using PNG for uploads that require color value retention. I have tested out a jpeg and discovered that I can retain color values with that file type, which I will use moving forward. 
Good to know about PDF exporting and the padlock button. Very helpful.
Much Appreciated -A

TIFF would be better, as it will preserve data more accurately.

Hi @asummers, thank you for providing the files @,,, requested and for the additional information...
The PNG file format doesn't support the CMYK colour space which is why you're not seeing consistent colour values when exporting your colour swatches as a PNG file. Can I ask the logic of using the PNG file format with CMYK colour as it wouldn't be the correct choice but understanding what you are trying to achieve would be really helpful and allow us to suggest alternative options for you...
The issue with your Designer file is that even though the default CMYK colour profile is set to U.S. Web Coated (SWOP) v2, your actual Designer document is using a Generic CMYK colour profile which you can see when you go to File > Document Setup and click on the Colour tab.
Exporting your CMYK document as a PNG file and then opening it in Photo results in all the colour swatches being converted RGB colour. In the colour palette if you unlock the padlock icon next to the colour space dropdown and then use the colour picker to click on a swatch you will see the colours in your file have all been converted to RGB values.
Exporting to PDF, if you export your file and tick the Embed Profile check box then your PDF will be exported using the same Generic CMYK Document Profile used by your Designer file so when opened in Photo using the settings shown in your screengrab the colour values will match.
If you export the Designer file as a PDF but don't tick the Embed Profile check box then you are exporting an unprofiled document which means when you open it in Photo it will be converted to the document profile set in your colour preferences, i.e., U.S. Web Coated (SWOP) v2 and that is why you are seeing the difference in the swatch colour values.

The PNG format does not support CMYK. It's always RGB.