After 2 years, they have probably filled that cooler, so the comb should be pretty sturdy. However, it is highly likely that they will have attached it to the lid. Not much you can do about that, unless you are willing to turn the cooler upside down and cut the bottom out to access the comb. Of course that will be very disruptive for the bees, and may break some comb in any case.
I suggest opening the lid a crack, then running a long blade along the underside of the cooler lid to cut through the comb. If you don’t have a blade long enough, you could try a length of foundation wire, fishing line, or even dental floss. You just want to separate it so that it doesn’t break as you rotate the lid open. If you don’t want to do that, you could unfasten the hinges, and try lifting the lid, but I suspect it will be too heavy with all of the comb attached, and a lot is going to fall off anyway. I think separating the comb from the lid will give you the best chance of getting it out in the largest possible pieces.
I think your bees are very lucky to have such a careful attentive keeper. Please keep us updated when you can, and photos would be wonderful if you are able to take them as you do the cutout. I wish you all the best!