Removing tubular tire
 

How do I do it without damaging the carbon rim?

No tools, both based on rim damage and on tire damage.

This works well for aluminum, I can't be sure pulling the glue to pieces won't affect the carbon fibers.

Hold the rim up in front of you with two hands, opposite the valve, and tire deflated. Push the tire sideways off the rim with both thumbs at the same point, trying to roll thetire off the rim, breaking the glue bead. It will take some effort to get this first area to loosen. Once it does, begin to work your way similarly around the wheel. When maybe 20 % of it is off sideways you can usually start to grip the tire better and just pull it off.

We need someone to chime in and say whether this technique is correct for carbon rims.

Road Fan

If the tire is no good, just cut across the tire, through the base tape, stick a finger in one open end and peel it off.

That sounds like a good idea, but I'm deathly afraid of cutting into the carbon...

Start the cut right next to the valve, push up on the valve and that should peal enough tire off the rim to be able to cut the tire without damaging the rim.