[clangd] Config: also propagate in sync (testing) mode

Summary:
I hit this while trying to add a config-over-LSP lit test, which I think
is an appropriate way to test this feature.

That needs a few more changes though...

Reviewers: kadircet

Subscribers: ilya-biryukov, javed.absar, MaskRay, jkorous, arphaman, usaxena95, cfe-commits

Tags: #clang

Differential Revision: https://reviews.llvm.org/D83802