What equipment can I used to dig a foundation very close to another structure?
So called 'mini-diggers' (as they are often called in the UK) now go down to a size which will easily fit through a 900mm doorway. Your situation isn't an impossibly tight dig. I'd recommend simply visiting your local tool/digger hire centre and hiring the largest machine you can get into the space you have. Don't forget you'll need room to turn the boom around at least 180 degrees. Possibly a micro, micro-plus or mini. 

NOTE: Terminology will vary from country to country depending on what excavator manufactures are present.