This can be done in Palmtree Panic Zone Act 1 with some Time Travel here and there as there are almost 700 rings in the level between the three eras. These do not need to be held all at once as I had 193 rings in Palmtree Panic Zone Act 1 and was hit. The achievement popped up as I was recollecting my rings.
Another way to do this is to find the angel statue in Wacky Workbench Zone Act 1 and touch it so that rings come out. After collecting them exit the room and re-enter as more rings will come out of it. Repeat until 200 have been obtained.