Unable to upgrade Cmake from 3.29.0 with Homebrew

I'm on macOS 10.14.6 Mojave and Xcode CLITools is in version 11.3.1.0.1.1576735732. Theese can't be updated. I have CMake installed with Homebrew under my home directory:

$ which cmake
/Users/vjorma/.brew/bin/cmake

$ cmake --version
cmake version 3.29.0

When I try

brew upgrade cmake

, I get

==> Upgrading cmake
  3.29.0 -> 3.29.3
Error: The contents of the SDKs in your Command Line Tools (CLT) installation do not match the SDK folder names.
A clean reinstall of Command Line Tools (CLT) should fix this.

Remove the broken installation before reinstalling:
  sudo rm -rf /Library/Developer/CommandLineTools

Install the Command Line Tools for Xcode 11.3.1 from:
  https://developer.apple.com/download/all/ 

I can do no such thing, I'm on a school computer with no administration privileges. Both CMake and the Xcode command line tools are working just fine, I use them everyday. I'd like to find a Hombrew-based solution and avoid building Cmake myself, because it takes so long.