For earlier Pro releases, we had made a custom build of cartopy and put that up into our Esri channel, but hadn't done that work for Pro 3.0. Because of its use of a number of packages which overlap with the core distribution of Pro, it may be challenging to get a solid build without some work on our end, but I would start with this:

Run a clone from the UI or with `conda create --clone arcgispro-py3 --name cartopy --pinned`. The `--pinned` here is important because that will constrain the package set for things like Notebooks to the versions we use in Pro. In this created environment after activating, try just running `conda install cartopy`. If that doesn't solve, it means that some of the packages Pro requires aren't compatible with the build up on Anaconda. As a short-term fix you may want to make a separate environment without ArcPy and install cartopy into that, but obviously that's not ideal.

Cheers, Shaun