You have privacy within ecash through Chaumian blind signatures, but you lose it if your mint has only one keyset/user as your Lighning anonset is 1 (only one funding LN tx that mints corresponding ecash)

so if you fund a mint, then settle LN invoices using the same mint, it is the same as you just paying LN invoices from your node directly

If you do a multimint swap and then pay the LN invoice from another mint you may gain *some* privacy, but a LN channel connects the two mints so theoretically it is the same privacy as routing payements on LN.

If privacy is the feature you are seeking, I would suggest using trusted mints with more users in it, or set one up witu a group of your trusted friends, and have many unique LN funding and invoice payements interacting with it...

but what do I know, I am just a cat on a keyboard 

 Makes sense, thank you kitty cat