I can’t help but think that I’m somehow missing the question that you are actually asking. An sRGB ICC profile already has the “D65” source white point incorporated into the profile by means of the chromatic adaptation that was used to make the sRGB ICC profile from the sRGB color space specs. It doesn’t have to be added again, in the context of using ICC profile color management.
I tried an experiment once, modifying LCMS to use D65 as the illuminant, instead of D50. I installed the modified LCMS in /usr (this is on Linux) so all ICC profile applications used this modified LCMS. And I made a set of ICC profiles that used D65 as the illuminant. With this modified LCMS, when making ICC profiles from color space specs, D65 color spaces didn’t need to be chromatically adapted to D50. But D50 color spaces such as ProPhotoRGB did need to be chromatically adapted to D65. And of course “E”, D60, and etc color spaces still needed to be chromatically adapted, but to D65 instead of D50.
When I tried editing using this modified LCMS and the modified profiles with my editing software such as GIMP, all the colors looked exactly the same. Exactly the same. The only way I could get “different colors” was to:
- Use my “D50 illuminant” ICC profiles with my modified-to-use-D65 version of LCMS
- Or else use my “D65” version of LCMS with my normal D50-adapted ICC profiles: Edit: What I meant to say, should have said, was “Use the D65-illuminant profiles with normal non-modified D50-illuminant LCMS”.
Sometimes you might run across an incorrectly-made sRGB ICC profile where the chromatic adaptation from D65 to D50 wasn’t done. Using such a profile makes images look blue, such as the image below on the right (the colors on the left are correct):
Back in the days of V2 workflows, you could “get different colors” - either blue or yellow or even other colors depending on your monitor’s actual white point, by using Absolute colorimetric rendering intent to the monitor.
You could also get different colors when converting from one ICC RGB working space to another ICC RGB working space with a different white point, but you had to specifically ask for Absolute colorimetric intent - all the editing software I’ve ever seen defaults to Perceptual or Relative, so nobody was likely to do this accidentally.
For example, you might convert from from sRGB to BetaRGB (which has a D50 white point) or vice versa, using Absolute colorimetric intent, resulting in images such as are shown below. Notice the image on the right is “too yellow” and the image on the left is “too blue”:
But the ICC decided this sort of color change when using Absolute colorimetric was confusing to users.
So for V4 workflows, when the source and destination color spaces are both “monitor class” profiles (all the standard RGB working spaces we use in the digital darkroom are “monitor class” profiles), when you ask for Absolute colorimetric rendering intent, what you get is Relative. Which makes it decidedly more difficult to write tutorials that encourage users to experiment and thus learn for themselves first-hand the difference between relative and absolute colorimetric intents 
The images above come from my article on “Will the real sRGB profile please stand up”, which was written when I actually had access to V2 editing software: https://ninedegreesbelow.com/photography/srgb-profile-comparison.html