In practice higher "colour temperature" LED bulbs will seem brighter for any given input power due to the way they are made. Usual construction appears to be a high efficiency blue LED emitter with phosphors to down convert some of the blue emission to other colours. Nominal red and yellow phosphors seem to be common in headlamp bulbs.
Higher colour temperature specification bulbs let more direct blue radiation through and divert less to the phosphors. There are significant energy losses in wavelength down conversion via phosphors so letting more blue through clearly means more light for the same input power. The blue wavelength is also quite close to the peak sensitivity of the rod cells in the eye. Rod cells are pure intensity detectors and play no part in colour vision. They are very sensitive indeed, almost single photon sensitive in ideal conditions, and do pretty much all the work when the eye is dark adapted. So not only do you get more light power out of higher colour temperature bulbs that power is where the dark adapted eye is most sensitive so things look even brighter at night than pure power calculations would suggest.
Applying colour temperature specifications to LED bulbs is scientifically fraudulent. Colour temperature applies only to incandescent light sources whose output over the appropriate spectral range is adequately close to that of true black body. For quantum physical reasons (Planck's Law) the spectral output power curve of a Black Body is the same shape whatever the temperature. The peak just gets higher and shifted to a shorter wavelength as it gets hotter. http://earthguide.ucsd.edu/eoc/teachers/t_universe/p_blackbody.html Basically just grab the peak of the graph and drag leftwards and upwards.
Can't find good picture of white LED spectra on t'net but plot b "W-LED" here https://www.researchgate.net/figure/237151257_Output-spectra-of-the-colored-light-sources-a-fluorescent-lamp-b-W-LED-c shows the general idea. A sharp blue peak with a hump lower down, mostly yellow & red. Colour temperature is assigned by finding a temperature at which the blue peak and yellow-red tail of the LED output pretty much touch the black body curve.
Really the whole thing only sort of works because of the human brains remarkable ability to interpret the eye output signal so colours and shadow intensities are seen pretty much consistently over a huge range of light levels and source colours. When it comes to apparent colour what you see is along way from what the eye got.