You can NOT. There is no workaround, there is no shortcut, there is not an alternative way.
You SHOULD change the color mode to CMYK, adjust the color and use a file format that accepts CMYK. Your options are TIF or PSD itself.
The CMYK value needs to include the color profiles used. Normally this would be "Adobe 1998" and one specific where you live, for example, SWOP, Fogra, Japan, etc.
It has no sense that the client asks a CMYK PNG, it does not matter if it is for a webpage. That is NOT how things work.
When you are reading a CMYK value from the color dropper on Photoshop you are NOT reading any CMYK value. The program is "inventing" those values on the fly in the case you want to actually convert it to CMYK and save it in a file format that supports it.
Edited to clarify @Danielillo's comment.
I need to assume somethings.
You want PNG format probably because you know it is a lossless format for a webpage. On a web page, you need an RGB image.
But in this case, the client feels insecure about using another color that another designer (wrongly) defined as CMYK, just because that is what the company's image design guidelines states.
A color conversion is done on a matrix of ranges. On an RGB file, you have potentially 16 million combinations of values to get potentially 16 million different colors.
But if you add another channel you have potentially 16 million x 256 or 4 billion combinations of colors...
But there are several issues here.
a) You can not get 4 billion colors from a 16 million source. You convert values from ranges.
b) On a CMYK print, you have even less color gamut than an RGB file so you actually have less than 16 million colors.
c) On CMYK file, regardless you are actually using 256 levels on each channel you are rounding the values from 0 to 100%. This is, the color combinations of the matrix use a maximum of 100 ^ 4 or 100x100x100x100 possible combinations, that again exceeds the real gamut of the color mode.
d) A color has different ways to be interpreted on a print. It is not very common but the two ways are a "chromatic combination" or an "achromatic" one.
A color can be defined as a combination of CMY+K inks or a combination of K+CMY ones... Yes I know that sounds silly.
This means when a color gets neutralized by a combination of C+M+Y values, this color can be replaced by a K ink.
e) A color profile does not shoot black ink on light colors, for example, to avoid the user seeing black dots.
All these elements give you "gaps or steps" on the color conversion.
On @Danielillos tests he assigned Y:80% but this value jumped to Y:79%
again because the RGB value was reinterpreted as that value.