For my part you are welcome, Happy coder.
But especially the very last bit of this sounds ominous:
"get sign-off from the client who will test the colours in their offices"
If they use what are usually called office printers, these have no PostScript (emulating) driver but print based on RGB internally converted to CMYK, so if you send a carefully developed CMYK document, the printer software will first convert it to RGB at its own discretion and then convert back to CMYK at its own discretion (again), in other words a double conversion beyond your control to something that may well be clearly different to what you have started out with. Obviously, prints from different such printers may turn out differently.
In that case I am afraid it will be better to send the RGB that is as close as possible to what is expected in print and let them see whether the print (that they prefer) is satisfactory, including being sufficiently close to the RGB they see on their monitors (the quality also being beyond your control) which may be different to what you see on yours.
Apart from all that, it is difficult to compare additive colours on screen and subtractive colours on paper, lighting being crucial.
Since the artwork has flat colors and doesn't appear to have any effects applied it should work fairly well in a CMYK to RGB conversion.
Matters are very different if any fills or effects using transparency are involved. Conversions from CMYK to RGB or vice versa might work okay if the blending mode is normal. If blending modes like multiply, overlay, etc are used then it is very likely serious color shifts would occur. These kinds of color shifts can be easily seen when copying artwork from a RGB-based Illustrator document and pasting them into a CMYK-based Illustrator document.