It sounds like you might be using the power meter incorrectly. The power meter has a wavelength setting, but this only affects the calibration. It does not change the operation of the sensor. The wavelength setting is used to compensate for differences in the response of the sensor so you can get an accurate reading. For instance, most sensors have a peak somewhere, and the response falls off on both sides of that peak, so the meter has to report a higher power for the same sensor output the further the wavelength is from the peak. Here is the responsivity curve for the S120C power sensor from ThorLabs, which uses a silicon photodiode and has a wavelength range of 400-1100 nm (https://www.thorlabs.com/images/tabimages/S120C_Responsivity_780.gif):
In your case, the LED is likely producing most of its output power around 623 nm. Shining that light on the sensor will result in the sensor emitting some amount of photocurrent. The meter then converts that photocurrent to optical power by looking at the responsitivity curve. If you select 400 nm instead of 623 nm, the responsivity of the sensor at that wavelength is lower, so the power meter will compensate for that and report a higher power. If you set the wavelength to 950 nm, you would get a lower power. If you set it to 1100 nm, you would get an even higher power.
So in your case, you aren't measuring the LED spectrum, you're measuring the power meter calibration curve. If you want to measure at different wavelengths, then you have to use an optical filter of some sort to select the wavelength that you're interested in.