Can I estimate an LED bulb’s color temperature from a photo?

Two possibilities:

1. You camera has a "Custom white balance" and will display the corresponding color temperature(*): take a picture of a gray chart or equivalent, apply custom white balance, check the result temperature.

2. You can set arbitrary color temperatures in the camera's WB: shoot a series of pictures of a grey chart over a range of temperature settings, check which picture is the most neutral (R=G=B).

Edit: testing method #2

It turns out that my camera will display RGB histograms, so there is no need to transfer the pictures to a PC to check things out, everything can be done with the camera display.

So, a quick test with my bed lamp, shooting my color reference, changing the color temperature (which is conveniently add to the display):

The green histogram doesn't move, but the red and blue one eventually get aligned at 2700K.

Coincidence? This is exactly what the lamp markings say:

Edit#2: further tests

Testing with:

• two lenses:
  • Canon 35mm IS STM (the one used above)
  • Sigma 17-70mm
• two fluorescent lamps:
  • the 2700K one used above
  • a "Cold daylight" one (no markings on the lamp, but rated 6700K on the box).

Findings:

• Both lenses produce the same TC readings
• Accuracy at the other end of the spectrum is worse, the camera has to be set to 7800K to align red and blue histograms.
• At the high end of the spectrum (8000K) a wide variation of the TC settings results in minute histogram shifts, while at the low end (2700K) a single step (100K) has a very visible effect, so the inaccuracy at the higher temperatures would not be that much of a problem, lamps with a 500k difference wouldn't make that much difference in the rendered colors.

(*) Not my Canon 70D, unfortunately