Jo,

you want consistent colors as Spot, in CMYK and in RGB.
The color is defined as Pantone Coated spot ink 2563C.
Right ?

1. Define in Photoshop your RGB space, for instance sRGB.
2. Define in Photoshop your CMYK space, for instance ISO Coated.
3. Define as well a Gray TRC, maybe Black Ink ISO Coated
(can be explained later, TRC=tone reproduction curve,
a better substitute for a single dot gain number).
4. Define as well a Spot TRC, maybe Black Ink ISO Coated.
5. Save these settings (any name)as sRGB-ISO-ISO-ISO.csf
6. Synchronize by CS# Bridge.

How to get the numbers:
Photoshop / Color Picker / Color Libraries / Pantone Coated 2563C.
Read the Lab numbers: L=70, a=24, b=-24. These are independent
of RGB and CMYK choices. Absolute references.
Back to Color Picker / Read CMYK values / Read RGB values /
Read hex values for RGB (used in HTML, practically only for sRGB).

Why is this workflow better than taking the numbers from a
printed or digital swatch book ?
Because the CMYK numbers are valid for a SPECIFIED printing
process (your choice), which contains inks, paper and dot gain.

How to check the conversion quality/robustness:
Fill a rectangle by 2563C or simply by the Lab color.
View / Proof colors for ISO Coated / Gamut warning: no problem.
View / Proof colors for sRGB / Gamut warning: no problem.

Result: this spot color can be printed by spot ink, by CMYK
ISO Coated and it can be reproduced nicely on monitors which
are near to sRGB.

If the percentages for Y and K (for the example 2563C) should
be very low , then I would try to replace these values by 0.
For ISO Coated these values are 0.

Hopefully I understood your task correctly.

Best regards --Gernot Hoffmann