Look for the CRI rating of bulbs that you buy. It's a measurement of how close to a blackbody spectrum the bulb is putting out, the highest fidelity being 100. Note that this is not the temperature measurement, and you can have e.g. 2700K or 5000K bulbs with high CRI.
Newer LED phosphors are typically 90+ CRI, and I commonly find 93 CRI bulbs available off the shelf.
I didn't want to mention that CRI is matched against the spectrum of _daylight_ because of the confusion that happens with color temperature when you mention the "daylight" word. You're right though, the CRI reference spectrum is matched against sunlight rather than a true blackbody.
Even high cri lights have a huge blue spike that doesn't match the sun. I don't know what chip OP uses, but you need a full spectrum light if you actually want very sun-like light. This page has some details:
Interesting. The Wikipedia entry mentions SPD and I think that is where I think LEDs fall down—having a skewed and/or incomplete spectrum. Even though it may make certain target colors look correct.
Newer LED phosphors are typically 90+ CRI, and I commonly find 93 CRI bulbs available off the shelf.
https://en.wikipedia.org/wiki/Color_rendering_index