How to protect constant LED current from ripples & clamps?
Working off of your schematic, you need to stabilize the supply to the opamp. Without a stable voltage to the opamp, you lose control of the "accuracy" of the opamp.

Take the 12V half sine waves, put in a reasonable amount of capacitance to limit the ripple. Using an estimate as the starting point -- 100Hz, 1V ripple, 1A current drain => 1A * 0.01 sec / 1V = 10mF capacitor. The capacitor is a little big, perhaps 5mF and tolerate 2V of ripple.

With the valley of the "rippled" supply being 11 or 10V, that places a limit of what voltage you can get with what linear regulator for a stable supply. Find a modern linear regulator to supply the opamp and voltage reference.

Choose a modern opamp (e.g. not 741) that matches the operating conditions. Now the current regulation circuit in the schematic should be able to give you good current regulation although Q1 perhaps should be a darlington (or MOSFET).

Also, to further reduce current ripple of the LED, put a capacitor directly across the LED.

I don't know what is the existing Voltage Reference Chip. Personally, I would not use a 7805 for example. Get a modern more accurate and stable regulator or use a TL431/TLV431. Also, with a 5V reference, that wastes 5V at the bottom and cuts out large chunk of headroom for the opamp circuit to operate. For example, a 1.25V-TLV431 works well with all kinds of modern 5V rail-to-rail opamps.