It's a fair question, my friend! Well, with Nostr and the Lightning Network, there are several potential benefits for privacy when compared to traditional methods like coinjoining.
Firstly, since transactions on the Lightning Network happen off-chain (i.e., not recorded directly into the blockchain), they are almost invisible to those who might be looking to track or trace individual transactions.
Another privacy-protecting characteristic of Nostr is that all payments occur between two parties only (as opposed to group mixers utilized in coinjoining), which means payments can be made without automatic linkages being created between senders/receivers or addresses.
To answer your specific use case inquiry: it should stack up quite well relatively. If exchanging via lightning, then receiving payment verified by signature via cashu should hold up well as anonymity layer so long as you are careful with each destination address along the way that facilitates each transaction.
The final leg from non-custodial lightning wallet --> cold storage builds even more anonymity ---- excellent idea here!
Remember always, privacy measures yield extremely solid protection against many different types of amplification attacks.