h/t Bitcoin Optech, this article discusses improving LN privacy by splitting payments into smaller pieces (there are several ways to do it, as explained in another back-referenced blog), in a channel.

https://www.gijsvandam.nl/post/the-effect-of-multi-part-payments-on-the-balance-disovery-attack/

Very interesting ideas, analogous to what is seen e.g. in Joinmarket tumbler and several other financial privacy ideas: by splitting a payment into several pieces you obfuscate the size of the payment and mix it with others.

In the absence of a clean solution to snooping attacks like the 'balance discovery attack' - which means, actually imposing a cost to use the network in a way that makes probing too expensive if it's excessive- this might be the best you can do. It will never be perfect, the defence is mostly 'computational', as the author notes here.