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Do what other said, use a blade and hammer it in between the carbon and crown race, this is one of the very rare instance that a simple method exceed those expensive tools as thpwe tools need to be in excellent condition to barely make a dent.
When you put a new one in, cut the crown race to split it and shove it in, this way you won’t need to cut it off again.
If the crown race use the same angle cartridge bearing as the new headset; leave it on.
It's not really going to be doable due to being too close to carbon.
Just choosing between these two designs.
The first one - icetools looks sturdier but I'm not sure it will fit in the spacing.
Second one looks thinner but reports are it's not very strong.
Does anybody have an icetoolz and can measure it accurately for me? I would really appreciate!