Dash Light question

I have a 1999 Toyota Camry LE 4Cyl with just a little over 200M miles on it.
The other day the little orange dash warning light came on that indicates a light is out on the rear end of the car.
Upon investigation, I discovered that in fact the rear window deck brake light was out as well as both of the license plate lights. I then replaced all three bulbs with new and much brighter LED bulbs which works beautifully.
Then, to reset the dash warning light, I did as I always have done in the past by disconnecting the negative battery cable and turning the ignition switch to the on position for 30 seconds or so. The next morning I started up the car and as I expected, the warning light was gone. However, as soon as I stepped on the brake pedal to shift to reverse I noticed the same warning light re-appeared. I then told my wife that I must be missing another burned out bulb somewhere on the car so between the two of us we checked and double checked but could not for the life of us find another burned out bulb. We checked all the signals, running lights, trunk lights, backing lights and found nothing.
I have tried manually cancelling this warning light again and again but it keeps re-appearing for some reason.
I notice when I first start the car this warning light is not there; and it stays off until I put my foot on the brake pedal to shift into reverse to back out of the garage. As soon as I touch the brake pedal the orange warning light suddenly appears and stays lit until I turn off the car; and then re-appears again the next day when I first touch the brake pedal.
All of my lights are working just fine, but this warning light continues to re-appear every day.
I appreciate your thoughts.
Thanks

My thoughts---LED lights draw a whole lot less amps then incandescent bulbs. So how do you think the car knows when a bulb is burnt out in a circuit? How about by measuring amps? in the individual circuits? I will bet that if you put OEM bulbs in the circuit, that all will be fine. But if you really like LEDs, then change them all, then buy a resistor kit, or a variable potentiometer, and wire it into the circuit and experimentally determine what value resistance shuts the light off on the dash.....Basically fooling the circuit into thinking it has incandescents, while you thrive in the LEDs.

Just my thoughts....

I think brcidd hit the nail on the head.