The LED voltage will change as it heats up, so you can use a regulated voltage PSU as long as you set the final adjustment when hot, otherwise current will rise as the LED heats up and its Vf drops.
I think it's better to use a constant current supply, you could use a 3A LM338 (the big brother to the LM317 adjustable regulator) and a resistor to set the current at 1.25v. However that will cost you some 2.5-3v overhead and lots of watts.
The best performance solution would be to set the regulated PSU's internal current limit at 3A, and just connect it to the LED and run it as a constant current DC supply. Depending which IC is in the PSU, it is probably as simple as one resistor inside the PSU that sets its current limit (assuming this is a standard offline SMPS regulated DC supply). A photo of the insides of the PSU would help determine if it can be modified.