My question is: I know the lightning network works differently, but would there be a way to "prove" I own a certain amount or even revealing the balance? If so what would be the process? Sure I could publish somewhere the balance but then again you would need to trust me to publish the right number.
Hmm. It's not foulproof but I guess you could publish all of the received individual donations in terms of invoices? If you exclude a certain invoice then that would be proveable by the individual who paid the invoice by publishing it?
Not sure what this would prove as it does not at all imply that you used the donated funds as intended, something that could much easier be tracked if you were to be using an on-chain address.
though here you could also most definitely probably publish the invoice of it going to charity xyz's node along with the preimage