What do you call a short burst? I take it youre using a star screwed down to the aluminum disc? Though you cant really measure LED temperature, with a good star, and good method of mounting, if you can keep your hand on the disc, then chances are the LED isnt too hot, provided you are driving it within normal specified currents. Just a SWAG(scientific wild *** guess) , you should be able to run for at least a minute or two on your heatsinnk, probably more.
If you want to get a worst case scenario idea of how long you can run a burst, assume no heat is removed from the disc, calculate or measure the weight of it, multiply by the specific heat of the aluminum and temperature difference to get your heat capacity, and see how many joules/calories/watt-hours it would take to heat the plate from your ambient temperature to 60C or so, and then assume 95% of the energy into the LED comes out as heat, (or 100% for real worst case scenario) and divide the heat capacity by the led's heat output to get a time. In practice it will take longer because heat will be dissipated from the heatsink via convection any any other air flow. (Radiation doesnt dissipate that much heat when your heatsink is only a few dozen degrees hotter then ambient! people get hung up on increasing radiant dissipation when they really shouldnt even worry about it)