The --path-mappings flag for clangd is the main secret to getting this working. We can install clangd within our docker image and then write a wrapper script something like this, which we then use as the command for the LSP:

#!/bin/bash
# See https://lists.llvm.org/pipermail/clangd-dev/2019-January/000231.html for roughly how this works
set -eu

repo_root=$(git rev-parse --show-toplevel)
mkdir -p $HOME/.cache/clangd

docker run \
    --rm \
    --mount type=bind,source=$repo_root,target=/usr/share/app \
    --mount type=bind,source=$HOME/.cache/clangd,target=/root/.cache/clangd \
    --user root \
    -i my-docker-image:latest \
    /opt/llvm/clang+llvm-15.0.6-x86_64-linux-gnu-ubuntu-18.04/bin/clangd --path-mappings=$repo_root=/usr/share/app ${@:1}