Lessig's theory of anonymous donations seems to have been carefully developed for the sole purpose of exonerating Ito, who is Lessig's friend. It's not like there's some new revelation that's come out since Lessig wrote his piece.
I think no theory of anonymity works. A villain isn't going to keep their pinky-promise to not tell everyone that your prestigious institution took their money.
It’s not particularly difficult to make a donation anonymous in a way that the recipient genuinely doesn’t know who made the donation. Just have a neutral third party handle it. This could be a donor advised fund or a trustee, for example. I suspect this isn’t particularly rare.
That would be part of handling anonymous donations sensibly. The transaction should be between the development office and whatever trustee or foundation gives the gift. The ultimate beneficiary shouldn’t be notified of the donation until, say, the end of the fiscal year, at which point all anonymous donations are aggregated.