How to measure room light without LEDs interfering?
Normally you should be able to switch LEDs off, make a measurement and switch LEDs back on without a noticeable delay. Brightness variation is noticeable below 100Hz, which translates to 10ms period. 

What you need is a photodiode with fast response time (under 0,1ms or so, though there are devices responding in nanosecond range). Phototransistors and LDRs are considerably slower.

LEDs response time is in single-digit microseconds even for white LEDs, and RGB LEDs are even faster. Below is a response of typical LEDs to a 200ns pulse:



Regular AnalogRead takes about 0.1ms (and, again there are faster alternatives), so you can easily achieve kHz measurement frequencies where brightness variation is completely unnoticeable.