If anyone is interested, I've created the LSP plugin here (eventually I'd like this plugin to become part of geany-plugins):
https://github.com/techee/geany-lsp
It requires the latest Geany from git master as some things had to be modified in Geany itself to support this kind of plugins. As far as I can tell, the plugin should be reasonably stable and bug-free (at least I'm not aware of any major issues).
I'm definitely interested in any feedback so if you run into some problems or have some suggestions, please let me know (by creating a new issue in the above repository).