Methodology isn't necessarily bad but I know what you mean. However it is important to know what notation you want to use and that leads to the question regarding how much detail you want.
ArchiMate is really aimed at Enterprise Architecture level so its not good for detailed level. More like a black box notation where you don't care whats in the box. For instance the business objects and data objects don't have attributes.
UML is fine for getting down to the nitty gritty code and database levels. So more like white box where you do care whats in the box.
From what I interpret from
We have a customer who has asked us to document their systems starting at a high level and then drilling down into them.
It sounds like its down to detailed code and database tables and attributes so UML is probably the best choice. Note that the elements can be represented by graphics to make them more ascetically pleasing to the eye for certain stakeholders.
Done some similar models and my solution architects include things like this;
System [Package]
Use Cases - to describe what it does
Components - to describe what its made up of
Deployment Diagram - to show the components deployed onto servers
Package and Class Diagrams - show the software architect and construction
State Models - to describe any object states and events
Interactions diagrams - to show any interactions at system/component/class levels
That usually covers what most people want. Haven't done any UML for around 11 years now so there might be some other stuff in the latest version of UML but the above works for us.