Though we are using LN directly and solely for payments this unpredictability problem is something we are most likely going to handle with zero fee routes between relays where payments go for relaying sessions.

I'm not 100% sure yet, precisely, but shortening paths by making 3 of 3 multisig channel opens and the triads broadcasting when the balance is substantially lower on one side than the other two.

Chaumian cash was my original, 10 years ago general concept, I called them "vouchers" at the time. But I think a lightly changed form of LN deployment is probably simpler (in terms of not having to build payment rails from scratch). We only need sender anonymity, not bidirectional.

Bidirectional anonymity is another matter, but really sits at a higher level of abstraction, as it can be achieved from the base of source routed onions (like LN does, but not variable length).