It's been a cold wet morning, so I've played around with some profiles. This first set is three profiles created using Adobe Profile Editor from the same test card shot under different lighting. The top one is mid-day summer daylight from last year, the middle is flash, and the bottom is a daylight balanced LED light (I accidentally put LCD on the comparison, but it's actually LED).
As you can see, there are some differences presumably caused by the different spectra of the different light sources (and perhaps slight exposure differences, although I tried to match them as close as possible). I usually shoot in daylight using the same daylight profile shown here. I never personally use flash, but if you are a regular flash shooter I think it would be best to create a separate profile purely for your flash shots, as the differences between the daylight and flash examples above seem to justify it.
Now here's the test card that was originally shot in daylight, with the daylight profile applied at the top and an averaged profile at the bottom. The averaged profile was created by averaging the values for each colour patch in the daylight, flash and LED profiles. (When you're creating custom .dcp profiles using APE, you only need to worry about the 24 patches on a standard Macbeth/ColorCheck card. The software then twists the colour space around those values automatically, and also extrapolates the tungsten profile from your 6500K original.)
It seems to me that just these three profiles combined averages out to something very close to the straight daylight profile on its own. I'd suggest that the more different artificial light sources you average out, the closer you'll get to something that's indistinguishable from daylight. In which case, if you want one general purpose profile, you might as well just shoot the test card in daylight.
And how does any of this apply in the real world? Here's a random shot converted from raw to jpeg using my normal daylight profile on top, and the averaged profile at the bottom. Even just averaging out three different light sources seems to me to result in a difference that's negligible in real-world photography. I'd expect it to be impossible to detect any difference at all if you averaged out enough different illuminants.
Ah well, that kept me happy for a couple of hours on a wet Saturday morning. The methodology was definitely more rough-and-ready than lab-grade, but I hope it gets the basic idea across.
(Note: all test cards were shot using the Pentax K-S1 and SMC Takumar 24mm/3.5, so the equipment used is the same in all examples.)