Yeah, good point with frying the meter!

Then what about having a few small shunt resistors (1, 5 and 10 ohmd or something like that) put paralel with the led one at a time instead of just the probes.
Then measure the voltage accross that and calculate the current as I = V/R. If the current matches all cases the assumption of a constant current source is valid and we have a current value.
If is not a cc source things can go bad for then circuit but with the info provided is all I can think of.

The resistors will all draw their own current and confuse the IC about what the load is. There is no viable solution to the problem and I feel you are being "pwned"
https://en.wikipedia.org/wiki/Pwn

 

Hahaha! That depends on the circuit and the missing info.
To explain a bit better let's assume that the Led is fed a constant current through a proper current source (that's the only hint we have so let's assume is valid) and then cathode connects to gnd.
Unlike voltage sources current will take the path of least resistance thus if there is a resistor in parallel with the led it means that on the anode terminal Va = I*R. Chosing the right resistance so Va is much less than the forward voltage of the led means the led is not conducting. At that point only lekage current passes from the led which will be really low compared to the source current.
Finally, since the majority of current will go through the resistor we have a good curent estimation.

Now, if there is a feedback circuitry around the led etc, that changes stuff and you might have a point (or pawing ).

 

There is an error in your thought process. Forcing a constant current into a low resistance produces a low voltage. For example 20 mA into 1Ω results in 20 mV across the resistor. Are you confident you can measure 20 mV with any particular degree of precision?

 

Sorry, I skipped this part.. But still, this is more like temporarily altering the board rather than breaking. After measuring, the led can be placed back like it was before.

But besides that, since the led is powered through a constant current source wouldn't mean that bypassing the led (with one of lower Vf or just multimeter probes across its terminals) and measuring the current passing through would still give a correct current value?

If you have a constant current source, then simply remove the LED, solder a 1x2 female header back in place and measure the current with the LED in series. Then you know exactly how much is going through the LED and likely the source. Or look at the components and figure it out.

 

If the LED is being driven from a constant current source then shorting the LED out should not change the current. So if you connect your digital multimeter set to a current range directly across the LED is should read the current that would be going through the LED. As the voltage across the test leads when set to a current range will be very low (Probably less than 100 mV.) there will be no significant current going though the LED with less than 100 mV across it. The current to voltage graph for a LED is very none linear. With 0.5 volts across a RED LED I could not see any current through it with my meter set to the uA range.

Les.