It's easier to work in CMYK and then convert to RGB than the other way around.
At least the CMYK gamut is present in the RGB one. The RGB one isn't reachable in CMYK most of the time. On top of this, all the different devices, platforms (gamma) and software might show the RGB color differently.
The client also need to accept that CMYK cannot be used for some projects and absolutely needs to be converted. But in general, the colors don't change much from CMYK to RGB. If your client is very picky with color-matching and doesn't like the converted CMYK then you have not many other options than adjusting them manually to tone down the colors a bit!
The issue you might encounter is if you use Pantones and let your InDesign or other software convert them to RGB automatically (or even CMYK.) If you don't need to have these Pantones in spot colors and only in CMYK, then convert them already in CMYK and remove the spot. This way the software won't match the spot Pantone name to a "pre-made" conversion.
Personally, I prefer to create a document for print and one for web, or at least not mix Pantones/CMYK together when not necessary; it's fast with software like Indesign and there's quick tricks to relink the images and adjust the styles. For example, moving the CMYK image folder and linking the RGB images from another folder, using the same file names (note: be sure to be careful when using this technique.) This way I have total control on the conversion and can adjust the colors manually if necessary. When I can't export in different color modes, I prioritize CMYK and then convert my pure CMYK file in RGB using Adobe Acrobat Pro with the Preflight tools. That's another quick way that doesn't require you to export the file twice.
If your client wants the nice flashy RGB blues or greens to fit the color library you want to use in CMYK/Print as well, then tell them it's impossible!
Convert your Pantones manually to get a closer matching CMYK/RGB.
Automatic software conversion or online values don't always match well. In the end, what you see is what you get. It's possible to get a CMYK and RGB value very close to the Pantone if you adjust your CMYK manually.
See the link below, it could work for you.
Mismatched CMYK Values