I'm not sure I understand the situation well enough, so apologies if this is useless.
In my opinion, strategy pattern is for things that are abstractly the same operation, but with variations that you want to choose at runtime. When I think of an API, the operations aren't usually interchangeable, so I would give them different (incompatible) types.
If you're asking whether you should split each 'operation' (API endpoint or otherwise) into it's own class, then probably yes imho. That's what I understood from the "...Strategy" postfix. That seems like something that'd make the growing of functionality more manageable.
Both ways seem like Command pattern to me (the API request being the command), but maybe I'm misunderstanding the intended use.
I feel like I'm extrapolating a lot in my interpretation, so feel free to disregard if not useful.
stuff ;)