One of our customers, frequently wants changes for the RTE style that is places in the appsettings.json. This always means that we have to build and deploy a new version of the Backoffice to update the RTE with the new style. Is there a way to place the string somewhere else, that does not required a redploy?

You can do a lot by changing them on the datatype: https://docs.umbraco.com/umbraco-cms/fundamentals/backoffice/property-editors/built-in-umbraco-property-editors/rich-text-editor/rte-styles

I don't know if this might also help https://umbraco.themediawizards.co.uk/the-grimoire/formatting-for-richtexteditor-config/