We use the plastic shims a lot with good results.
Only like the translucent plastic ones because they are quite hard. The white ones are too soft and allow rocking.
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Back on the subject, the process we use: Dry set the toilet on the shims to see how much you need to get to where you want to be. It sound like it will take more than one shim, but I would be cautious if you use two, because of the huge gap. Tape the location of the shims, remove the bowl. Leave the shims in place then set the wax and toilet. After it is tightened down, use a wood chisel and a hammer to cut flush with the china (assuming there is some plastic showing, we then typically use clear silicone, but in your case if the gap is big, maybe white would be better.
White silicone is unforgiving, so mask the floor first. Good Luck.