'I think I'd go for "Zighder Zay"'

I agree there's probably no real solution for an English speaker to tackle the 'ui' sound. for 'Zee' I'd suggest to aim more for an 'a' sound for a more general Dutch feel (while still stopping well short of aiming for ABN), rather than tack on the 'y' although, as you suggest, that would turn up, for example, in a heavy Amsterdam accent.