This method is empirical, there must be a better way out there.
Without a datasheet where the manufacturer did the empirical experiments for you, there is no other way.
A better way is to use an LED IV analyzer.
I use my multimeter in diode mode to get Vf. Here I got 2.4V.
You need two multimeters to do this. One to provide the current (in ohms mode) the other to measure the voltage.
If the Vf is indeed 2.4V (not likely) it must be a red, orange, or yellow LED. More likely your meter uses 2.4V to test. LEDs have a higher Vf than other types of diodes.
set the current limit to ~400mA
immediately check the current draw (here 270mA) and set the current
limit to that value.
This is puzzling. With this procedure the current should have gone to 400 mA and stayed there. Why did it only go to 270 mA? What limited it to 270 mA? Possibly the Vf is more than the voltage your were using.
If you are indeed using a CC/CV driver then the voltage should not matter, it will adjust itself. However you do want to start out with a lower current (e.g. 300 mA or less).
This is a waste of time.
1. You should not buy LEDs that do not have a datasheet.
2. How much will knowing the forward voltage help? Hint: Not much.
Even if you have a datasheet there is not parameter in the datasheet for your PCB and its thermal management. The datasheet will not say what the temperature is for any given current.
Either way, datasheet or not, you must not allow the LED to get too hot. How hot is hot? Check the datasheet. Get the max Tjunction and the junction to case thermal resistance calculate the thermal resistance for your PCB and heatsink to estimate the temperature. This is the simplest app note I know of to help you do that: Thermal Design By Insight, Not Hindsight
Use the low tech empirical method I use. No Vf required.
Touch the PCB near the LED.
If it burns, the temperature is too high.
I prefer a temperature so that I can put my finger on the PCB for an extended amount of time without much, if any, discomfort.