Hi @Henr1k,
as you already found out, managing updateble synonyms in-line in the index settings is not possible at the moment and there are no real plans to do so in the future. Instead, we hope to make managing synonyms easier with a dedicated API (see https://github.com/elastic/elasticsearch/issues/38523), however current efforts with that regard have been stalled for a while.
You can see how to upload resources like synonym files to Cloud here I believe: https://github.com/elastic/elasticsearch/issues/38523, however after reading it I'm not 100% sure if uploading a new synonym file as a bundle like described there will make it visible to already running nodes or if there is an alternative cloud API to do that. I would suggest also asking in the cloud forum around here about that since I'm not so knowledgeable around that topic tbh.
Regards